Basically, you can do it with the standard Java API. Check out URL, URLConnection, and maybe HttpURLConnection. They are in package java.net.
As to the API specific signature, try sStringToHMACMD5 found in here.
And remember to CHANGE YOUR API KEY, this is very IMPORTANT, since everyone knows it know.
String payload = "{\"method\": \"addUserFavoriteSong\", ....}";
String key = ""; // Your api key.
String sig = sStringToHMACMD5(payload, key);
URL url = new URL("http://api.grooveshark.com/ws3.php?sig=" + sig);
URLConnection connection = url.openConnection();
connection.setDoInput(true);
connection.setDoOutput(true);
connection.connect();
OutputStream os = connection.getOutputStream();
PrintWriter pw = new PrintWriter(new OutputStreamWriter(os));
pw.write(payload);
pw.close();
InputStream is = connection.getInputStream();
BufferedReader reader = new BufferedReader(new InputStreamReader(is));
String line = null;
StringBuffer sb = new StringBuffer();
while ((line = reader.readLine()) != null) {
sb.append(line);
}
is.close();
String response = sb.toString();