Where can an adult learn to backflip?

You could also try calling http://www.gypsygeoff.com/Milwaukee_Community_Circus/Home.html. If they can't teach you I'm sure they could point you in the right direction.

http://www.gymnastike.org/page/Adult-Gymnastics

You need a gymnastics studio or club (not a 'gym' but actual gymnastics). Most of the people who run these studios will be happy to do extra one-on-one work, especially if they own the space. Failing this, back flips are a part of certain modern dances, such as street dancing, so modern dance studios or classes will present similar opportunities. (I know my daughter's street dancing teacher will do things like this, for example.)

Yup, a gymnastics club would help. At 6', 195 lbs, it's safe to assume that you should be able to pick up the skill pretty quickly. You don't sound very overweight or too underweight to generate the muscular power required to do it.

Thirding gymnastics. If you are in the US, look for a local gymnastics gym that has a boys' team because they are more likely to have a coach that's capable of spotting you and the equipment to let you practice safely. If you PM me your location (if you're in the US), I can help you find your local gyms that have a boys' team. Once you find a gym, just call and you should be able to set up some private lessons.

Another option would be a parkour training facility - they're popping up in some places and would certainly do the job.

4th here. I am almost exactly your size and build although I haven't done a backflip in 10 years. I have little doubt I could do one again after a brush up with a gymnastics coach. You can learn fairly quickly if you have decent balance and leg strength and a spring floor but a coach will probably have you work on a series of drills before they're comfortable with you putting your head over your ass without a spot.

Most capoeira schools will have some backflippers around, and would be more casual than gymnasts as far as your having that singular goal.