Build your reputation. Start out by competing in (and winning) smaller competitions in and near your local area. You can find these competitions by simply asking others involved in your particular sport.
Contact businesses to try to obtain a sponsor. Shoot for a business that is at least somewhat well-known. The local mom and pop hardware store or deli won’t cut it. You will need to make a strong case for why the business should sponsor you. A highlight reel and competition titles will go along way.
Enter a Winter X Games qualifying event. You can find a list of qualifying events through ESPN, the company that arranges and controls the X Games. You can also peruse publications related to your particular sport (such as skiing or snowboarding).
Win or place high in the qualifying event to be allowed admission to compete in the Winter X Games. The rules will be outlined prior to starting. For example, a competition might allow the top three snowboarders to qualify. You may need to compete in several rounds of qualifying depending on the event.