Is it possible to make a JFrame that has a transparent background and draw an image on it, so only the image will be visible with no border or background?
Yes, it's possible in many ways. This is one of them:
setUndecorated(true); setBackground(new Color(1.0f,1.0f,1.0f,0.5f));
4th float (which I set to 0.5f) in Color's constructor is alpha channel. It can be 0.0f - 1.0f depend on transparency you want.