Try Dawn Original -Put soap on the rag and use circular motion -let set for a about 10 minutes -wipe off with a wet rag. If your doors get as bad as mine this may take a couple times to get it all.But this has always worked for me. I have done this on job sites because 9 times out of 10 no matter how good the paint is it will not cover the grime.It bleeds through even after 3 and 4 coats. There is no cure for preventing it-I now wipe down door jambs and other fingerprint spots after I finish dishes at the end of the day