I had old cabinets that had several coats of paint and I used a heat gun. Kind of like a hair dryer, but heated a lot hotter. The paint bubbled up and I scraped it off with a thingy like a putty knife. Came off pretty good. You can also use that chemical paint remover too.
If you want to do it cheap, buy some Citristip at your home center, it is much less obnoxious than traditional MEK paint stripper. Apply in the morning, reapply at midday, and strip in the afternoon. More than one attempt may be necessary. Citristip also has a wash to remove it from the surface. This all very labor intensive but cheap to do yourself.A much easier alternative to to take the doors to a professional stripper who will dip them and return slick as a whistle. Not cheap though.