Qualifications
You've already got your PhD, right?
The qualifications are to be funny. It really doesn't matter how many classes you've taken, what books you've read on the art of stand-up, or even who you know. You just have to get up on that stage, stand behind that mic, and make people laugh until they're purple. If you can do that on a regular basis, you may actually be able to move up in the world of comedy.
Otherwise, you'll probably have to settle for being a joke. Or an accountant. We hear that's a decent gig, too.