It is possible to paint melamine, but it is important to properly prepare the surface. Start by cleaning the melamine with TSP, then sand it with 80 or 100 grit sandpaper. Then, carefully dust off the surface before applying two coats of Zinsser BIN. Once this step is complete, you can apply two coats of General Finishes Milk Paint or Fusion Mineral Paint. If desired, you can finish with a varnish of your choice. 

For even stronger adhesion, you can use Fusion's Ultra Grip product directly on the melamine or on other glossy, hard surfaces. This product creates a strong adhesion for paint.