If you are already familiar with servlet you do not need much to create a simple server to achieve what you want. But I would like to emphasize that your needs will likely to increase rapidly and therefore you may need to move to a RESTful framework (e.g.: Spring WS, Apache CXF) down the road.
You need to register URIs and get parameters using the standard servlet technology. Maybe you can start here: http://docs.oracle.com/cd/E13222_01/wls/docs92/webapp/configureservlet.html
Next, you need a JSON provider and serialize (aka marshall) it in JSON format. I recommend JACKSON. Take a look at this tutorial:
http://www.sivalabs.in/2011/03/json-processing-using-jackson-java-json.html
Finally, your code will look similar to this:
public class Func1Servlet extends HttpServlet {
  public void doGet(HttpServletRequest req, HttpServletResponse resp)
    throws ServletException, IOException {
    String p1 = req.getParameter("param1");
    String p2 = req.getParameter("param2");
    // Do Something with the params
    ResponseJSON resultJSON = new ResponseJSON();
    resultJSON.setProperty1(yourPropert1);
    resultJSON.setProperty2(yourPropert2);
    // Convert your JSON object into JSON string
    Writer strWriter = new StringWriter();
    mapper.writeValue(strWriter, resultJSON);
    String resultString = strWriter.toString();
    resp.setContentType("application/json");
    out.println(resultString );
  }
}
Map URLs in your web.xml:
<servlet>
  <servlet-name>func1Servlet</servlet-name>
  <servlet-class>myservlets.func1servlet</servlet-class>
</servlet>
<servlet-mapping>
  <servlet-name>func1Servlet</servlet-name>
  <url-pattern>/func1/*</url-pattern>
</servlet-mapping>
Keep in mind this is a pseudo-code. There are lots you can do to enhance it, adding some utility classes, etc...
Nevertheless, as your project grows your need for a more comprehensive framework becomes more evident.