My first freelance gig came from oDesk a couple of years ago, and since then I've gone on to freelance full time. I'm not an expert on the subject by any stretch, but this is the best advice I can give:

* Don't try and compete on price. A lot of freelancers on the site can go lower, and you'll end up broke. Besides, if a client is more interested in saving a few dollars than paying for a decent worker, you're saving yourself a lot of future pain.

* When you apply for something, say what you'll do and give a rough idea of how you'll solve their problem. It only has to be a few paragraphs, but you'll have put more thought in than 99% of other applicants.

* Look for jobs that mention an ongoing relationship. This usually means they have a lot more work and want to try things out.

All of my work now comes from referrals which in some way originate from the first job I did on oDesk. It was discouraging when I was rejected for jobs despite putting in the effort, but the work paid off in the end. Best of luck!