Bummer your local school hasn't been meeting your needs.
I think with private school- first start with what their mission and philosophy are and how their program supports both. This will give you an idea of their program and focus.
For example- the school my DD (1st grade) attends- their mission is: "integrates high expectations for achievement, creativity, self-discipline and social responsibility. We cultivate in each student a lifelong enthusiasm for learning and a desire to contribute constructively to society. Our program provides students with an excellent academic, artistic, athletic and ethical foundation".
This is demonstrated in how their curriculum plays out- focusing on the academics but also integrating Art & Music into the curriculum. The 'social responsibility' comes through with each grade having a community service connection that they work on through out the year. Athletics is a focus as demonstrated by the whole school (K-8) doing XC Skiing twice a week for the entire winter in addition to other sports in the other seasons.
Matching your child to the school is as important as you finding what you want from the school too.
Also consider class size. One would hope that going to a private school would allow for smaller classes than what you would typically find in a public school.
Another question is where kids go after they complete all the grades of this school- and even further with success in college.
I don't know what your financial situation is, but don't be deterred by the cost- many private schools have financial aid available. My DD's school has about 30% of the population receiving some kind of aid.
Good luck with your search! It is really exciting when you find a school that you and your child really get excited about.