Exactly what must a student accomplish during their High School years that will result in a letter from Princeton signifying acceptance?
Short answer: There is no “magic formula”.
To be considered competitive, maintain high GPA, high standardized test scores, pursue ECs of your passion & interests, get amazing letters of recommendation, and win multiples of (national/international) awards.
Even then, you may not be accepted. So in short: do the very best you can.
Read this and replace every instance of “MIT” with “Princeton”: http://mitadmissions.org/blogs/entry/applying_sideways
There is almost nothing that can guarantee an acceptance from a school like Princeton short of maybe having your family donate a building, being an Olympian with a 4.0 GPA and the like. I suggest you do the best you can in every department (ex. grades, class rigor, standardized tests, ECs you are truly invested in, recommendations, essay), Don’t set your heart on one hyper-competitive school when there are so many great colleges and universities out there – be sure to apply to a realistic range of schools.
Oh come on, [everybody knows exactly what Princeton wants…](Favorite ECs of Different Top colleges? - #12 by marvin100 - Applying to College - College Confidential Forums)
Huh?