Tap on your Player Profile in the top left corner of the main screen.

In the Profile, you can change your name as well as your avatar.

How do I change my Profile name?

Tap your current Profile name, type your new Profile name, and then tap the Confirm button.

How do I change my avatar?

To select or change your game avatar, tap on your Player Profile in the upper left hand corner, press it again, and then choose your preferred Champion!