Here i am posting complete code for drawing route between current location and another place on google map.
For more details : Visit draw route on google map with current location  
PointParser class
import android.content.Context;
import android.graphics.Color;
import android.os.AsyncTask;
import android.util.Log;
import com.google.android.gms.maps.model.LatLng;
import com.google.android.gms.maps.model.PolylineOptions;
import org.json.JSONObject;
import java.util.ArrayList;
import java.util.HashMap;
import java.util.List;
public class PointsParser extends AsyncTask<String, Integer, List<List<HashMap<String, String>>>> {
    TaskLoadedCallback taskCallback;
    String directionMode = "driving";
    public PointsParser(Context mContext, String directionMode) {
        this.taskCallback = (TaskLoadedCallback) mContext;
        this.directionMode = directionMode;
    }
    // Parsing the data in non-ui thread
    @Override
    protected List<List<HashMap<String, String>>> doInBackground(String... jsonData) {
        JSONObject jObject;
        List<List<HashMap<String, String>>> routes = null;
        try {
            jObject = new JSONObject(jsonData[0]);
            Log.d("mylog", jsonData[0].toString());
            DataParser parser = new DataParser();
            Log.d("mylog", parser.toString());
            // Starts parsing data
            routes = parser.parse(jObject);
            Log.d("mylog", "Executing routes");
            Log.d("mylog", routes.toString());
        } catch (Exception e) {
            Log.d("mylog", e.toString());
            e.printStackTrace();
        }
        return routes;
    }
    // Executes in UI thread, after the parsing process
    @Override
    protected void onPostExecute(List<List<HashMap<String, String>>> result) {
        ArrayList<LatLng> points;
        PolylineOptions lineOptions = null;
        // Traversing through all the routes
        for (int i = 0; i < result.size(); i++) {
            points = new ArrayList<>();
            lineOptions = new PolylineOptions();
            // Fetching i-th route
            List<HashMap<String, String>> path = result.get(i);
            // Fetching all the points in i-th route
            for (int j = 0; j < path.size(); j++) {
                HashMap<String, String> point = path.get(j);
                double lat = Double.parseDouble(point.get("lat"));
                double lng = Double.parseDouble(point.get("lng"));
                LatLng position = new LatLng(lat, lng);
                points.add(position);
            }
            // Adding all the points in the route to LineOptions
            lineOptions.addAll(points);
            if (directionMode.equalsIgnoreCase("walking")) {
                lineOptions.width(10);
                lineOptions.color(Color.MAGENTA);
            } else {
                lineOptions.width(20);
                lineOptions.color(Color.RED);
            }
            Log.d("mylog", "onPostExecute lineoptions decoded");
        }
        // Drawing polyline in the Google Map for the i-th route
        if (lineOptions != null) {
            //mMap.addPolyline(lineOptions);
            taskCallback.onTaskDone(lineOptions);
        } else {
            Log.d("mylog", "without Polylines drawn");
        }
    }
}
FetchURL class
import android.content.Context;
import android.os.AsyncTask;
import android.util.Log;
import java.io.BufferedReader;
import java.io.IOException;
import java.io.InputStream;
import java.io.InputStreamReader;
import java.net.HttpURLConnection;
import java.net.URL;
public class FetchURL extends AsyncTask<String, Void, String> {
    Context mContext;
    String directionMode = "driving";
    public FetchURL(Context mContext) {
        this.mContext = mContext;
    }
    @Override
    protected String doInBackground(String... strings) {
        // For storing data from web service
        String data = "";
        directionMode = strings[1];
        try {
            // Fetching the data from web service
            data = downloadUrl(strings[0]);
            Log.d("mylog", "Background task data " + data.toString());
        } catch (Exception e) {
            Log.d("Background Task", e.toString());
        }
        return data;
    }
    @Override
    protected void onPostExecute(String s) {
        super.onPostExecute(s);
        PointsParser parserTask = new PointsParser(mContext, directionMode);
        // Invokes the thread for parsing the JSON data
        parserTask.execute(s);
    }
    private String downloadUrl(String strUrl) throws IOException {
        String data = "";
        InputStream iStream = null;
        HttpURLConnection urlConnection = null;
        try {
            URL url = new URL(strUrl);
            // Creating an http connection to communicate with url
            urlConnection = (HttpURLConnection) url.openConnection();
            // Connecting to url
            urlConnection.connect();
            // Reading data from url
            iStream = urlConnection.getInputStream();
            BufferedReader br = new BufferedReader(new InputStreamReader(iStream));
            StringBuffer sb = new StringBuffer();
            String line = "";
            while ((line = br.readLine()) != null) {
                sb.append(line);
            }
            data = sb.toString();
            Log.d("mylog", "Downloaded URL: " + data.toString());
            br.close();
        } catch (Exception e) {
            Log.d("mylog", "Exception downloading URL: " + e.toString());
        } finally {
            iStream.close();
            urlConnection.disconnect();
        }
        return data;
    }
}
Main Activity
public class MainActivity extends AppCompatActivity implements GoogleApiClient.ConnectionCallbacks, GoogleApiClient.OnConnectionFailedListener, com.google.android.gms.location.LocationListener , OnMapReadyCallback, TaskLoadedCallback{
   //variables for map and route
    private GoogleMap mMap;
    private MarkerOptions place1, place2;
    Button getDirection;
    private Polyline currentPolyline;
    private MapFragment mapFragment;
    private boolean isFirstTime = true;
//variables for current location
    private static final String TAG = "MainActivity";
    private TextView tvLocation;
    private GoogleApiClient mGoogleApiClient;
    private Location mLocation;
    private LocationRequest mLocationRequest;
    private com.google.android.gms.location.LocationListener listener;
    private long UPDATE_INTERVAL = 2 * 1000;  /* 10 secs */
    private long FASTEST_INTERVAL = 2000; /* 2 sec */
    @Override
    protected void onCreate(Bundle savedInstanceState) {
        super.onCreate(savedInstanceState);
        setContentView(R.layout.activity_main);
        //code for getting current location
        requestMultiplePermissions();
        tvLocation = (TextView) findViewById((R.id.tv));
        mGoogleApiClient = new GoogleApiClient.Builder(this)
                .addConnectionCallbacks(this)
                .addOnConnectionFailedListener(this)
                .addApi(LocationServices.API)
                .build();
    }
    //code for drawing route
    @Override
    public void onMapReady(GoogleMap googleMap) {
        mMap = googleMap;
        mMap.clear();
        Log.d("mylog", "Added Markers");
        mMap.addMarker(place1);
        mMap.addMarker(place2);
        CameraPosition googlePlex = CameraPosition.builder()
                .target(new LatLng(22.7739,71.6673))
                .zoom(7)
                .bearing(0)
                .tilt(45)
                .build();
        mMap.animateCamera(CameraUpdateFactory.newCameraPosition(googlePlex), 5000, null);
    }
    private String getUrl(LatLng origin, LatLng dest, String directionMode) {
        // Origin of route
        String str_origin = "origin=" + origin.latitude + "," + origin.longitude;
        // Destination of route
        String str_dest = "destination=" + dest.latitude + "," + dest.longitude;
        // Mode
        String mode = "mode=" + directionMode;
        // Building the parameters to the web service
        String parameters = str_origin + "&" + str_dest + "&" + mode;
        // Output format
        String output = "json";
        // Building the url to the web service
        String url = "https://maps.googleapis.com/maps/api/directions/" + output + "?" + parameters + "&key=" + getString(R.string.google_maps_key);
        return url;
    }
    @Override
    public void onTaskDone(Object... values) {
        if (currentPolyline != null)
            currentPolyline.remove();
        currentPolyline = mMap.addPolyline((PolylineOptions) values[0]);
    }
    //runtime permission method
    private void  requestMultiplePermissions(){
        Dexter.withActivity(this)
                .withPermissions(
                        Manifest.permission.ACCESS_FINE_LOCATION,
                        Manifest.permission.ACCESS_COARSE_LOCATION )
                .withListener(new MultiplePermissionsListener() {
                    @Override
                    public void onPermissionsChecked(MultiplePermissionsReport report) {
                        // check if all permissions are granted
                        if (report.areAllPermissionsGranted()) {
                            Toast.makeText(getApplicationContext(), "All permissions are granted by user!", Toast.LENGTH_SHORT).show();
                        }
                        // check for permanent denial of any permission
                        if (report.isAnyPermissionPermanentlyDenied()) {
                            // show alert dialog navigating to Settings
                            openSettingsDialog();
                        }
                    }
                    @Override
                    public void onPermissionRationaleShouldBeShown(List<PermissionRequest> permissions, PermissionToken token) {
                        token.continuePermissionRequest();
                    }
                }).
                withErrorListener(new PermissionRequestErrorListener() {
                    @Override
                    public void onError(DexterError error) {
                        Toast.makeText(getApplicationContext(), "Some Error! ", Toast.LENGTH_SHORT).show();
                    }
                })
                .onSameThread()
                .check();
    }
    private void openSettingsDialog() {
        AlertDialog.Builder builder = new AlertDialog.Builder(MainActivity.this);
        builder.setTitle("Required Permissions");
        builder.setMessage("This app require permission to use awesome feature. Grant them in app settings.");
        builder.setPositiveButton("Take Me To SETTINGS", new DialogInterface.OnClickListener() {
            @Override
            public void onClick(DialogInterface dialog, int which) {
                dialog.cancel();
                Intent intent = new Intent(Settings.ACTION_APPLICATION_DETAILS_SETTINGS);
                Uri uri = Uri.fromParts("package", getPackageName(), null);
                intent.setData(uri);
                startActivityForResult(intent, 101);
            }
        });
        builder.setNegativeButton("Cancel", new DialogInterface.OnClickListener() {
            @Override
            public void onClick(DialogInterface dialog, int which) {
                dialog.cancel();
            }
        });
        builder.show();
    }
   //methods for getting current location
    @Override
    public void onConnected(Bundle bundle) {
        if (ActivityCompat.checkSelfPermission(this, Manifest.permission.ACCESS_FINE_LOCATION) != PackageManager.PERMISSION_GRANTED && ActivityCompat.checkSelfPermission(this, Manifest.permission.ACCESS_COARSE_LOCATION) != PackageManager.PERMISSION_GRANTED) {
            // TODO: Consider calling
            //    ActivityCompat#requestPermissions
            // here to request the missing permissions, and then overriding
            //   public void onRequestPermissionsResult(int requestCode, String[] permissions,
            //                                          int[] grantResults)
            // to handle the case where the user grants the permission. See the documentation
            // for ActivityCompat#requestPermissions for more details.
            return;
        }
        startLocationUpdates();
        mLocation = LocationServices.FusedLocationApi.getLastLocation(mGoogleApiClient);
        if(mLocation == null){
            startLocationUpdates();
        }
        if (mLocation != null) {
            // mLatitudeTextView.setText(String.valueOf(mLocation.getLatitude()));
            //mLongitudeTextView.setText(String.valueOf(mLocation.getLongitude()));
        } else {
            Toast.makeText(this, "Location not Detected", Toast.LENGTH_SHORT).show();
        }
    }
    @Override
    public void onConnectionSuspended(int i) {
        Log.i(TAG, "Connection Suspended");
        mGoogleApiClient.connect();
    }
    @Override
    public void onConnectionFailed(ConnectionResult connectionResult) {
        Log.i(TAG, "Connection failed. Error: " + connectionResult.getErrorCode());
    }
    @Override
    protected void onStart() {
        super.onStart();
        if (mGoogleApiClient != null) {
            mGoogleApiClient.connect();
        }
    }
    @Override
    protected void onStop() {
        super.onStop();
        if (mGoogleApiClient.isConnected()) {
            mGoogleApiClient.disconnect();
        }
    }
    protected void startLocationUpdates() {
        // Create the location request
        mLocationRequest = LocationRequest.create()
                .setPriority(LocationRequest.PRIORITY_HIGH_ACCURACY)
                .setInterval(UPDATE_INTERVAL)
                .setFastestInterval(FASTEST_INTERVAL);
        // Request location updates
        if (ActivityCompat.checkSelfPermission(this, Manifest.permission.ACCESS_FINE_LOCATION) != PackageManager.PERMISSION_GRANTED && ActivityCompat.checkSelfPermission(this, Manifest.permission.ACCESS_COARSE_LOCATION) != PackageManager.PERMISSION_GRANTED) {
            // TODO: Consider calling
            //    ActivityCompat#requestPermissions
            // here to request the missing permissions, and then overriding
            //   public void onRequestPermissionsResult(int requestCode, String[] permissions,
            //                                          int[] grantResults)
            // to handle the case where the user grants the permission. See the documentation
            // for ActivityCompat#requestPermissions for more details.
            return;
        }
        LocationServices.FusedLocationApi.requestLocationUpdates(mGoogleApiClient,
                mLocationRequest, this);
        Log.d("reque", "--->>>>");
    }
    @Override
    public void onLocationChanged(Location location) {
        String msg = "Updated Location: " +
                Double.toString(location.getLatitude()) + "," +
                Double.toString(location.getLongitude());
        tvLocation.setText(String.valueOf(location.getLatitude() +"    "+String.valueOf(location.getLongitude())));
        Toast.makeText(this, msg, Toast.LENGTH_SHORT).show();
        if(isFirstTime){
            //code to draw path on map
            getDirection = findViewById(R.id.btnGetDirection);
            getDirection.setOnClickListener(new View.OnClickListener() {
                @Override
                public void onClick(View view) {
                    new FetchURL(MainActivity.this).execute(getUrl(place1.getPosition(), place2.getPosition(), "driving"), "driving");
                }
            });
            place1 = new MarkerOptions().position(new LatLng(location.getLatitude(), location.getLongitude())).title("Location 1");
            place2 = new MarkerOptions().position(new LatLng(20.8880, 70.4012)).title("Location 2");
            mapFragment = (MapFragment) getFragmentManager().findFragmentById(R.id.mapNearBy);
            mapFragment.getMapAsync(this);
            isFirstTime = false;
        }
     }
}
If you have any error then check out original tutorial given at the starting of this answer.