Post a link to your contest entry on your Facebook wall, and ask all of your friends to help you. If all of your Facebook friends aren't interested in sweepstakes, you can use Facebook Friend Lists to post your vote requests only to the friends you want to ask to vote for you.
About.com Contests' Sweepstakes Forum has a section devoted to exchanging votes. Post your vote request there, and be sure to return the favor by voting for the people who voted for you as well. Be sure to read the Vote Request Rules before getting started, to ensure that your post won't be removed.
If you don't know how to get started, see How to Create Facebook Groups from About.com's Personal Web Guide for tips.
Before you start soliciting votes on Facebook and elsewhere, keep in mind that not all contests allow their entrants to participate in vote exchanges. Be sure to read the rules carefully, to ensure that you won't be disqualified for participating in a vote exchange. Other contests may have restrictions on who you can ask to vote for you. For example, you might be allowed to ask friends to vote but not strangers, or you are allowed to post a link to the contest, but not to your specific entry.
If you are unsure, ask the sponsor for guidance about how vote requests can be made before you start.
For ideas on how to get votes outside of Facebook, see How to Get Votes for Contest Entries.