Be sure to read the submissions requirements for the contest. Pay close attention to specifics including contest rules, deadlines and instructions. Many entries may be disqualified when deadlines are missed or writers don't adhere to the rules.
Don't forget to mail the payment with your entry. Most contests require a submission fee for processing the entries.
Try to enter contests that are local and have less entrants. This way you selection may have a better chance of winning.
Compare your work with other writers who write in the same subject and that are considered really good in their category.
Join a writing community to communicate with other writers and receive an outside opinion on your work. Find others to critique your work, especially those who read a lot.
Where to Find Writing Contests Online:
http://www.writersdigest.com/competitions -Writer's Digests sponsors several competitions annually. Including poetry, fiction, non-fiction, self-published, and more!
http://www.freelancewriting.com/writing-contests.php - List of current writing contests in a variety of writing styles and genres.
http://www.writerswrite.com/classads/writingcontests/ - Writer's Write Writing contests.
http://www.writersweekly.com - Good Resource Writing Web Site
http://www.sfwa.org/beware/contests.html - Warnings on fake writing contents