Practice singing. No matter what else you do, it’s important to be able to carry a tune and sing on pitch. While you need not be the "best" singer at the competition, and you certainly don’t need to be professionally trained, those things won’t hurt you. Karaoke contests are usually judged on some combination of vocal ability, audience response, and stage presence, but if you can nail the singing part, the judges may give you some leeway on other aspects of your performance. The best way to practice karaoke is to sing karaoke.
Learn your songs by heart. Most karaoke contests allow you to pick your own songs, and if you’re going to win, you should have practiced these time and time again. Yes, the words are right there on the screen, but if you know the words and have perfect timing, you don’t even have to look at the screen. Not only will your overall performance be more masterful, you may impress the judges and audience with your knowledge of the songs. An elimination round in a contest is a bad time to find out that you can't pull off a particular song.
Understand the format of the contest. Karaoke contests are usually either judged by a panel of judges or by the audience. In some of the latter, the audience will actually formally vote, while in most a judge or judges will try to gauge the audience's reaction to a song. You also want to know how you're being judged. Most contests are based on overall performance, but some look strictly at how well you impersonate the artist (or how original you are), or other factors. Finally, know how many songs you'll be expected to sing so you'll have enough prepared in advance.
Follow the rules. Some contests require that you sing a particular kind of music (80s rock only, for example), or that you appear in costume. Make sure to follow the rules or you probably won't win even if you give the best performance.
Extend courtesy to other performers. Yes, other performers are your competition, but don't try to malign them or mock them, even if someone really can't sing. Be polite, and unless the rules dictate otherwise, applaud everybody. Remember you will very likely be judged on your audience's reaction to your songs, and many of your competitors are part of the audience. They won't treat a heckler kindly.
Cater to the audience (or the judges). Above all, sing to the audience, not to the karaoke screen. Beyond that, know your audience. Understanding your audience will help you choose songs that most appeal to them. If you haven't been to a venue before, you might want to visit it in advance to check out the crowd and see what they're playing on the jukebox. On the night of the competition, try to gauge the judges' or audience's reaction to songs and see if you can spot trends (maybe they hate love songs), and choose your songs accordingly if you still can.
Sing a song to showcase your range and talents. There are a lot of songs that just about anybody can sing passably, but if you've got an exceptional vocal range or if you can rap just like Snoop (rap is notoriously difficult to karaoke) choose songs that reflect that. If you want to awe the audience--and you do--you need a great performance of a difficult song.
Develop stage presence. People come to karaoke (or to concerts) to see a performance, not just to hear one. Don't just stand there and sing, and don't act like you don't know what to do during an instrumental interlude. Convey the emotions of the song with your facial gestures and body movements, and by all means dance if appropriate. Get some ideas by watching concert videos of professional performers. If you can find a video of the original artist performing the song you're going to sing, all the better.
Dress the part. Your attire can help the judges remember you, especially if you go to a themed competition or if you're only singing certain kinds of songs. If you're going to sing glam rock, dress like a glam rocker, or maybe dress like a country singer to add a bit of humor to your performance.
Be true to the original. Even if it's subconscious, most people judge a karaoke performance by how close it sounds to the original. It's amazing when someone gets up to sing a Garth Brooks song and you could swear they were just lip-syncing to a CD of Garth himself. While you need not be a professional voice impersonator, try to fit the mood and style of the song (i.e. add a bit of twang to a country song).
Add your own twist. Musicians who cover a song can alter it all they want. You don't have that luxury because the karaoke music is designed to sound just like the original version. That said, there's a little room for creativity. For example, try inserting the name of a local landmark in place of one mentioned in the song. This can have a humorous effect and is likely to get a rise out of the audience.
Try to go close to last in the rotation. You may not want to actually go last, because people tend to lose focus a bit in anticipation of the results or the next round. However, if you can position yourself near the end, the judges will have your performance fresh in their minds.
Relax and have a good time. If you have a problem with stage fright, you've got to learn to suppress it. There are a number of ways to do this, but probably the best is to fight it by singing a lot of karaoke. Don't take yourself too seriously up there, and even if you're nervous, show the audience that there's nothing you'd rather be doing.
Be a good winner (or loser). If you win, congratulations! Now be gracious. If you don't win, don't mope, don't make excuses, and don't get mad about how "the contest was fixed." You'll likely see these people again if you continue to compete in karaoke contests, and you want to leave them with a good impression of you.