Read the requirements carefully: some scholarships are awarded by nomination, while others require an application.
Begin your application in the fall. While you might submit the application at the last minute, taking time to assemble a complete, polished application increases the likelihood of success.
Submit a complete application, providing everything listed, such as transcripts, statements, and letters of support. If you put the time into an application, make it worthwhile; it's disappointing for us to set a candidate aside simply because of missing documents.
Ask for letters of support at least one month in advance. Your instructors need time to write a strong recommendation. When you ask an instructor for a letter, clearly state the deadline and what the scholarship uses as criteria for award. Your best strategy: meet with your instructor first.
Save all documents using your last name. Today, many judges prefer electronic copies of applications, and using your name in the title keeps your materials in order.
If the award requires your written work, and you are using essays and papers turned in for a class, use the professor's feedback to revise: it's meant to help you improve your writing, so use it to your advantage.
Proofread your work! Your application reflects your best efforts, and you don't want to compromise your submission with a misspelling like "litterature" (and it has happened!).
If you don't win this year, don't despair: within another academic year, you might provide the winning application.