I'm not too sure how to go about getting the external IP address of the machine as a computer outside of a network would see it.

My following IPAddress class only gets the local IP address of the machine.

public class IPAddress {

    private InetAddress thisIp;

    private String thisIpAddress;

    private void setIpAdd() {
        try {
            InetAddress thisIp = InetAddress.getLocalHost();
            thisIpAddress = thisIp.getHostAddress().toString();
        } catch (Exception e) {
        }
    }

    protected String getIpAddress() {
        setIpAdd();
        return thisIpAddress;
    }
}

I am not sure if you can grab that IP from code that runs on the local machine.

You can however build code that runs on a website, say in JSP, and then use something that returns the IP of where the request came from:

request.getRemoteAddr()

Or simply use already-existing services that do this, then parse the answer from the service to find out the IP.

Use a webservice like AWS and others

import java.net.*;
import java.io.*;

URL whatismyip = new URL("http://checkip.amazonaws.com");
BufferedReader in = new BufferedReader(new InputStreamReader(
                whatismyip.openStream()));

String ip = in.readLine(); //you get the IP as a String
System.out.println(ip);

One of the comments by @stivlo deserves to be an answer:

You can use the Amazon service http://checkip.amazonaws.com

import java.io.BufferedReader;
import java.io.IOException;
import java.io.InputStreamReader;
import java.net.URL;

public class IpChecker {

    public static String getIp() throws Exception {
        URL whatismyip = new URL("http://checkip.amazonaws.com");
        BufferedReader in = null;
        try {
            in = new BufferedReader(new InputStreamReader(
                    whatismyip.openStream()));
            String ip = in.readLine();
            return ip;
        } finally {
            if (in != null) {
                try {
                    in.close();
                } catch (IOException e) {
                    e.printStackTrace();
                }
            }
        }
    }
}

The truth is: 'you can't' in the sense that you posed the question. NAT happens outside of the protocol. There is no way for your machine's kernel to know how your NAT box is mapping from external to internal IP addresses. Other answers here offer tricks involving methods of talking to outside web sites.