Show that you are the best of the best.
First follow the tutorial on How To Post Audio.
When your profile shows the track you uploaded, click on the rightmost tab that looks like a trophy.
Click on the background image of the contest you want to enter. A popup will ask you to confirm the competition price and select one of your tracks to enter. Once the dropdown has been filled, click Proceed.
You will need to pay the competition fee with Paypal or your balance. An alert will appear telling you the transaction was successful once completed.
You will be redirected to the settings screen. Scroll to the bottom of the settings page, and look for a confirmation with a battle number and your track number. Congrats and good luck. You are submitted!