As a reward for their skill, Champions are granted many benefits in BFS. Champions are promoted to the Champion rank in the BFS Fanclub, and get their name mentioned in the description of the game and on the Champion display in the lobby. Champions also wield a special "Champion" name title in game, and when the 2nd half of the Rebirth update is released, champions will get the Champion Mysteliate.
Champion status is only for the most elite BFS players. In order to attain it, you must complete a ridiculously difficult task. This will be done in Boss Rush, and you will be tested against bosses. These can be Soul T, Crisis SpringWrath, and Sir Kadee. You do not have to fully solo these bosses, but prove that you are good enough at your class to attain Champion.
Ultimately, it's SupernalNine's decision on whether or not you're worthy of becoming a champion. Champion requirements may get crazier and more difficult as time progresses.