This is ver simple! All you need to do is earn trophies! You earn trophies by staying alive in battle longer than your opponents.
You can track your League progress simply by tapping on the League bar located at the top left of the main screen. You can claim your League Rewards here too! Also, if you scroll along you can see how you compare to your friends!
If you want to know more about Leagues, just tap here. :)