Start with having a solid academics background. Then top it off with a variety of extracurricular activities. Most effective if the activities show case your talents, interests, or passion.
Private schools are more expensive, but they also have more money to dole out than state schools. Look into all criteria for certain scholarships for your schools of interest. Some school will give out full rides from general funds (in my case) or they will give out full rides from a named scholarship program (like Heinz scholars program, in my friend's case). You need to be aware of all criteria and apply to each program if necessary.
Lastly, the trick is to go where you stand out. Schools value diversity, and being a minority, I looked at schools with high academic standards but less diversity. My diversity becomes something I can offer to the school in return for the scholarship.