Remove the doors from the cabinets and take off the hardware. This would include hinges and door knobs. Wash down the surface of the doors with a grease-cutting dishwashing soap and hot water.
Slide your fingers along the edges of each door feeling for a loose piece of thermofoil. If you can insert your fingers between the vinyl cover and the board, pull up on the thermofoil and peel it off. If loose, it should come off just like contact paper. Repeat this process until you are able to take off as much thermofoil as possible.
Sand down all the doors with a medium to heavy-grit sanding block or use an electric sander. Do this whether you remove the thermofoil from the door or not. If the cabinet still has the vinyl finish, sanding will roughen it up and allow you to paint over the surface. Wipe down the doors after sanding to remove dust and debris.
Apply a thick, oil-based primer to the doors. Follow the instructions on the product for proper application and drying times.
Fill in any holes in the door surface with putty. Add another coat of primer after the putty cures and you sand it flush to the door surface. Once the second coat of primer dries, paint the cabinet doors as desired.