I asked a similar question last year, here is what I found most helpful
What phonics program is used to teach early reading skills?
What math program is used & what topics will be covered?
What "specials" will they have... ie. art, music, PE, computers, library, etc & how long for each?
Recess time?
Behavior policy... ie. what system is used to reinforce good behavior & address poor behavior?
How much homework, type of homework? (yes we have homework in Kindergarten)
Units covered throughout the year?
I basically wanted an overview of the year & skills I would see addressed
Please make sure EVERY teacher is CURRENTLY state certified, especially if it is a religious school where they set their own rules. (Private schools are not required to have currently state certified teachers.)
Ask what methods they use to teach reading and what programs they use?
Ask maximum teacher-student ratio and if they might add a second class.
Stress that you want your child to be in the experienced teacher's class.
If your child has any special needs, really consider a school that will have the resources to help. Private schools are not the best place for children who need specialized education unless they have the trained staff for it.
And then also make sure you have a tour and get the feel for the school. That was the final deciding factor for us. The principal met with me and showed me the school and introduced me to various teachers and showed me most of the classes in progress, not just the Kindergarten class. We also got the opportunity to speak with parents of other children who had been attending the school.