You need to win seven matches to advance from the first league "Asteroid" to the second "Jupiter". Each win moves you along by a percentage (14 or 15%).
You play online league matches by going into online battle > league battle.
You don't lose any progress if you lose a game.
Obviously if you're struggling to win and have both a PS4 and a Vita then the boosting method described in the guide for "Smoke Me A Kipper, I'll Be Back For Breakfast" trophy will be the easiest way to get this trophy.