You are now able to fully customize your profile. To do this, go to 'Profile' in the top right hand corner of the game. Or, tap on your avatar in the main menu.
You can edit your name, change your avatar, change your country flag and choose alternative frames. Unlock new avatars and frames by unlocking Leagues and competing in Ranked Tournaments.
Aside from customization, you can also view your stats here, which include your favorite Legends and Champions, as well as your win rates in respective game modes.