[hardwired]

Components with a "J" prefix are Swing vis–a–vis AWT. In Swing we don't override the update method for graphics; its use is restricted to AWT graphics/drawing.
The preferred/recommended way to arrange for keyboard input to Swing components is with key binding. Using a KeyListener is okay but can cause focus–related problems.

Code:

import java.awt.*;
import java.awt.event.*;
import java.awt.image.BufferedImage;
import java.io.*;
import java.net.URL;
import javax.imageio.ImageIO;
import javax.swing.*;

public class GO extends JFrame {
    final int SCREENWIDTH = 700;
    final int SCREENHEIGHT = 700;
    boolean RESIZABLE=false;

    public GO() {
        super("click, click");
        this.setDefaultCloseOperation(JFrame.EXIT_ON_CLOSE);
        this.setSize(SCREENWIDTH,SCREENHEIGHT);
        this.setResizable(RESIZABLE);
        JPanel panela = new Vsebina(SCREENWIDTH, SCREENHEIGHT);
        this.setLayout(null);
        panela.setBounds(0,0,SCREENWIDTH,SCREENHEIGHT);
        this.add(panela);
        setVisible(true);
    }

    public static void main(String[] args) {
        new GO();
    }
}

class Vsebina extends JPanel implements Runnable {
    BufferedImage backBuffer;
    BufferedImage ozadje;
    Thread thread;
    boolean running = false;
    int x = 100;
    int y = 100;
    double theta = 0;
    double thetaInc = Math.toRadians(5);

    Vsebina(int w, int h){
        backBuffer = new BufferedImage(w, h, BufferedImage.TYPE_INT_RGB);
        Graphics2D g2d = backBuffer.createGraphics();
        g2d.setBackground(Color.pink);
        g2d.clearRect(0,0,w,h);
        g2d.setPaint(Color.red);
        int dia = Math.min(w,h)/3;
        g2d.fillOval(w/2-dia/2, h/2-dia/2, dia, dia);
        g2d.dispose();
        ozadje = loadImage(//"img/testOzadje.jpg");
                           "images/Bird.gif");
        registerKey();
        this.addMouseListener(ml);
    }

    protected void paintComponent(Graphics g){
        g.drawImage(backBuffer,0,0,this);
        g.drawImage(ozadje, x, y, this);
    }

    public void run() {
        while(running) {
            theta += thetaInc;
            int w = getWidth();
            int h = getHeight();
            double dia = Math.min(w, h)/4.0;
            x = (int)(w/2 + dia*Math.cos(theta)) - ozadje.getWidth()/2;
            y = (int)(h/2 + dia*Math.sin(theta)) - ozadje.getHeight()/2;
            repaint();
            try {
                Thread.sleep(50);
            } catch(InterruptedException ex) {
                stop();
            }
        }
        System.out.println("dela");
    }

    private BufferedImage loadImage(String path){
        BufferedImage image = null;
        try{    
            File temp = new File(path);
            URL tempURL = temp.toURL();
            if (tempURL==null) throw new Exception("sss");
            image = ImageIO.read(temp);  // or, read(file);
        } catch(IOException e) {
            System.out.println("read error: " + e.getMessage());
        }catch(Exception ex){
            System.err.print("NAPAKA pri branju slike");
            System.exit(-1);
        }
        return image;
    }

    private void registerKey() {
        int c = JComponent.WHEN_IN_FOCUSED_WINDOW;
        getInputMap().put(KeyStroke.getKeyStroke("ESCAPE"), "ESCAPE");
        getActionMap().put("ESCAPE", new AbstractAction() {
            public void actionPerformed(ActionEvent e) {
                System.exit(0);
            }
        });
    }

    private MouseListener ml = new MouseAdapter() {
        public void mousePressed(MouseEvent e) {
            if(e.getClickCount() == 2) {
                theta = 0;
            } else {
                if(running)
                    stop();
                else
                    start();
            }
        }
    };

    private void start() {
        if(!running) {
            running = true;
            thread = new Thread(this);
            thread.setPriority(Thread.NORM_PRIORITY);
            thread.start();
        }
    }

    private void stop() {
        running = false;
        if(thread != null)
            thread.interrupt();
        thread = null;
    }
}