Just like you built up experience in your field, you'll need to build up experience in freelancing. But that certainly can be done.
here you go!
Give away the secret: in your bid, explain to the client how you'll implement / build / program whatever it is they're looking for. If it's a Twitter application, let them know why the want to use the streaming API instead of the REST API. Give them enough information that they feel like they should use your response to evaluate the rest.
Be quick to respond: face it, on the freelancing sites a public posting will get a lot of responses. Use whatever tools are available to be one of the first responses the client sees. In the past, I'd use an RSS reader to track job feeds on oDesk.
Bring your own clients: do you have clients outside these online marketplaces / platforms? Transition them to billing through the platform you like best. That's an easy way bring a local reputation online.
Find the right jobs: this is doubly important for oDesk, where you're limited to a fixed number of bids per week. Setup searches that either identify jobs that match your experience well, or jobs that generally have less bids and you're able / willing to put in the work so you become experienced enough to respond to those jobs in a knowledgeable way. (Again, oDesk search + RSS feeds can be great here.)
Avoid the bottom: it may be tempting (all the while frustrating) to try to be the lowest bid. Don't do that. Yes, if you're starting out with no reputation as a contractor, you may need to bid lower than you'd want - but don't get caught up in trying to compete on price. This isn't a reverse eBay where the lowest bid wins - there are plenty of clients that ignore low bids, because they understand that reflects at least something about quality. Those are the clients you want.
The first three are what I did to establish my reputation. Anymore I rarely bid on jobs, generally I just respond to requests (called interview invitations on oDesk). Sill, when I respond I try to give the potential client a detailed response that illustrates I am the contractor they want to use.
The fourth is what I wish I could have done, but I started out my freelancing career online.