1. Make sure that your credentials are impeccable. You will probably get looked at for that English position if your degree is in American Language and Literature from one of the better colleges such as Brown, Duke, Stanford or Cornell. If your degree is from Harvard, Yale or Princeton, go to the top of the list. A bachelor's degree is ok, but you better make the right noises during the interview about your plans to finish your master's. Private schools vastly prefer to say that 80% of their faculty have advanced degrees. If your degree is terminal, that's cool too, as long as you are not terminal too.
4. Apply according to their explicit instructions. Follow application instructions to the letter. If for some reason known only to the headmaster, the school wants applications submitted on 3x5 cards, then so be it. In lieu of any precise instructions, submit a one page cover letter of interest together with your resume which should not be more than 2 pages and must include 3 references. Supporting documents such as transcripts and certificates can be submitted at a later stage.
5. Demonstrate your love of teaching. Private schools are all about excellent teachers. Parents pay a small fortune to expose their children's minds to gifted teachers. You must make sure that your love of teaching dominates any and all conversations with your future employer. It is really the only thing which matters.