Decide on a school. Each school has both its benefits (well-known, good curriculum) and its drawbacks (price, difficulty of acceptance), so you will need to size up each school and make a realistic guess at your chances of getting admitted into each one.
Take the SSAT/ISEE. This will be a requirement of almost any private school. The SSAT and the ISEE are standardized tests that these schools use to judge a student's abilities and potential. The better you do on this test, the more likely you will be to get into the private school of your choice.
Write a great essay. This will be part of the SSAT or ISEE, but you need to really focus on this part since the school administrator will read it directly. While the rest of the test will be scored and those scores will be sent to the school's administration, the actual essay will not be scored and will be sent to the school as well for review.
Make a donation to the school. While this isn't a guaranteed way to get accepted into the private school of your choice, it could help. Private schools receive no public funding and often look kindly on those who make generous donations. This is an option that should only be considered if you can afford it, since it may or may not help your chances of getting accepted.