1. Take cabinet doors off cabinets and remove all hardware.
   
2. Clean thoroughly with dry TSP (from your paint store) in warm water.  (Be sure to wear rubber gloves to avoid cracking the skin on your hands. )
   
3. Rinse Thoroughly.
   
4. Dry overnight. Next morning do a touch test. The surface should be dry and a bit rough. There should be no grease or  chalky residue. If the doors are still greasy repeat step 2.  If the doors have a chalk residue re-rinse them.
   
5. Paint the cabinets with Z prime or Kilz primer. This will keep any grease that you couldn't remove from bleeding through the paint.
   
6. Let dry for 2-4 hours.
   
7. Cover with 2 coats of latex kitchen and bathroom semi-gloss or gloss. Let dry overnight between coats. Allow 2nd coat to dry at least one full day before re-hanging the doors.
   

Note: The longer you allow the 2nd coat to cure/dry before re-hanging the doors the stronger it will be.

Spray paint your doors if possible.  You can rent the machine from most rental companies.  Lay down newspapers on a clean, vacuumed floor or table in a building (like your garage) where the doors will be protected while drying.  Support the doors on small blocks to keep the edges off the surface of the floor or table.  Consider the overspray you will get while painting.  You can staple clear plastic drop cloths to the ceiling of your garage and tape them together to form a temporary spray booth.  Make sure you have plenty of ventilation and you wear a respirator.  Make sure there are no pilot lights or open flames in the area (like water heaters or room heaters).  Have plenty of paint thinner/cleaner and rags on hand before you start.  Wear rubber gloves.

If you decide to paint by hand with a brush, use the correct brush and paint in the order indicated in the illustration above.

Numbered steps to hand painting a kitchen cabinet door with a paint brush.