Ugh. That's ass.

Acetone will take off the paint. It's particularly important to take it off mechanical parts — the rims (as they're the braking surface) and gears are most important. Acetone will attack non-paint plastics, as well, so you're probably best off taking the pulleys out of the rear derailleur.

This would be best done by stripping down the bike, then taking out plastic parts, then washing things off with acetone and a brush. Be sure to relube when you're done! Anywhere where metal touches metal should be greased or oiled.

If you can spare $25, a new chain will save you a lot of cleaning time, and it's a part you just replace every so often anyway.