• Make an image
• Write to the image, repaint
• Paint image to screen
• Override update to prevent flicker

Image img = null;
Graphics img_g = null;
public void init() {
    img = createImage(size().width, size().height);
    img_g = img.getGraphics();
    img_g.setColor(Color.black);
}
public void update(Graphics g) {
    paint(g);
}
public void paint(Graphics g) {
    g.drawImage(img,0,0,this);
}
public boolean mouseDrag(Event e, int x, int y) {
    img_g.drawLine(lastx,lasty,x,y);
    repaint();
    ...
}

Source