Some things to remember:
Ice time is limited, especially at the Learn To Skate/Basic Skills levels. Many rinks have restrictions on lessons during public skate, plus public skate in the winter is so crowded that lessons can be problematic. Be willing to work with the coach to find an acceptable time, which might include early morning, or skating at a different rink. Be open to it.
Know the cost. Private lessons include the coach's fee and the ice fee. The ice fee is the price of admission, essentially; what it costs to step on the ice. For public skating this will be anywhere from $3 to $7 depending on the rink. For "practice ice" (non-recreational ice set aside for lessons and practice), the fee will be higher, $6 to $20 depending on various factors. If you've got a Learn to Skate student, you'll be looking for cheaper ice and cheaper lessons. A high competitive coach will charge upwards of $40 or $50 for a half-hour lesson. Nothing wrong with hiring this coach if the match feels right, but you might want to think about other coaches who charge less (seldom lower than $25 per half hour). You can also ask about sharing lessons. I'll take group lessons as long as the students are no more than 4 years apart in age and/or span 3 or fewer skill levels.
High test and impressive competitive credentials are not the best indicator of good teaching. Conversely, a coach's record with students is a good indication. A coach who gets kids through high tests and whose kids do well in competition (relative to how much they practice) is someone to consider. If you're not looking for a so-called competitive coach, take this metric to the class level-- is this coach getting kids through class tests? Are her or his classes well organized and fun?
Look for coaches with students at your child's skill level (not ability, but skill level). If all of a coach's students seem to be at a much higher level, either that coach is pushing his/her kids through the levels too fast, or doesn't really work well with lower level students. Some coaches will tell you that they only take freestyle students. Good for them. (Works the other way 'round, as well. For instance, I only take students from beginner through Freestyle 2-3.)
You wouldn't obsess or worry about a violin or piano teacher, and it's really not that much different.