Look for the Three "C's" of Personal Fitness Training
Credentials
The most respected certifying bodies in the industry are the American College of Sports Medicine (ACSM), the American Council on Exercise (ACE), the National Strength and Conditioning Association (NCSA) and the National Academy of Sports Medicine (NASM). A masters degree in either Exercise Physiology or Kinesiology is a great academic foundation, but those who have an advanced degree should also have plenty of practical experience since the work is very much hands-on and a good student doesn't always make the best teacher.
Credentials should also include the ability to produce current quality references from clients, peers and medical professionals. And there's no substitute for five or more years experience. I also feel that as a general rule, it's more difficult to cultivate a strong business as an independent, rather than in a gym setting, since your success is completely dependent on repeat clients and growth from referrals, rather than a constantly overturning population of new gym members.
Competence
Besides the endorsements mentioned above, your trainer should be able to support the rationale behind his or her training approach with authoritative and unbiased sources of exercise and nutrition science. I provide all my clients with a comprehensive list of websites that I use as a foundation for my training philosophy including the ACSM website and the American Dietetic Association website, and I do my own regular voluntary continuing ed (besides the 20 hours every two years my certifying organization requires to re-certify) and make darn sure the direction I provide my clients is in accordance with the guidelines of those resources.
Chemistry
Ultimately, it's in the best interest of both the client and the trainer for the partnership to be a good fit. That means philosophical compatibility and clear, comfortable communication. It also doesn't hurt for there to be a good natural rapport, since training and lifestyle changes can be a sensitive, intimate issue. A free no-obligation consultation is a terrific opportunity to get a sense of that.
Of course location, rates and other logistics are important, and all that can be discussed either by phone in advance of a consultation or at the consultation itself.
One last thing - if you feel like the trainer is trying to "sell" you (interest you in training that doesn't feel right to you, persuade you to buy a big block of sessions up front at a discount or pressure you in any way at all) - run!! (Just make sure you don't have any knee, hip or ankle injuries before you break into your sprint! In that case, maybe you just want to cough loudly and wander away pretending you got an important call on your cell phone!)