Actually, Ness is incorrect. This is a convention where you're showing off a costume, not a play - DO show your back to the audience. The judges want to see that the back of your costume looks nice, and what if you have some special details there? Let them see it!
Now, I can't speak for contests where it doesn't matter what you're wearing, you could perform in a paper bag and it would be the same as performing in costume. Those contests are strange to me. Where I compete and judge, performance/presentation is weighted equally with workmanship and both are important to the competition. So the advice I can give is good for well-rounded performances and not skit-only competitions.
Have audio. Even if you're not doing dialogue, at least perform with background music. Walking around the stage to dead silence is a surefire way to be boring.
Use the emcee if you're allowed to - have them read an intro or read a punchline after you exit stage if you can. Don't make that intro a 10 minute long paragraph on your character's background, a single sentence to set the scene or lead in to your performance is enough. You don't HAVE to, but if you're stumped for how to get started, an emcee intro usually helps.
You don't have to do a comedic skit with dialogue. There are TONS of different ways to present your costume and character, choose something unique and appropriate. If you're wearing something beautiful, a choreographed piece of performance or pantomime to play out a scene on stage, over the right background music, is much better than trying to do a joke skit. Think about dancing, or at the very least, showing off. If you're wearing a fighter's costume, think about doing fight poses. If you have multiple people in your group, choreograph to move around each other in interesting ways that make the audience's eyes follow you, and show all sides of your costumes. Does your costume do anything neat like light up or transform? Show that off.
Use all of the stage, don't just stand in one spot.
Consider set pieces - a backdrop, a prop, a little sign, anything you can have them set on stage for you to give you a new entrance point, a point of interest, something to interact with, etc.
If you really want to be great, don't be satisfied with simply walk, pose, walk to new point, pose, walk off. That's the most basic of basics. Push yourself to do something neat. Punch it up. TELL A STORY. Use your movement, your music, and your costume itself to tell a nice 1 minute long story that the audience can understand even if they don't know your character.
Finally, don't feel like you have to use all of the time they give you. Going on too long is guaranteed to bore the audience.
Founder and President, Madison Area Costuming Society, a chapter of the International Costumer's Guild
Strange Land Costuming - www.strangelandcostumes.com