Go to your Profile screen. In the top-right corner of the profile card you'll see a pencil icon — tap it to open the Edit profile dialog. This icon controls all profile editing; there's nothing to tap on the avatar itself.
Inside the dialog you'll find three things: a photo control, a Username field, and an Email field. Email is shown for reference only — it can't be changed from this screen.
In the Edit profile dialog, tap the small camera icon overlaid on your avatar to open your device's file picker. The dialog's caption tells you what to expect: PNG or JPG, up to 5MB.
There's no separate display name in Crappin' Ain't Easy — your Username is what everyone sees, shown as @username. To change it, type a new value into the Username field in the Edit profile dialog and save.
Have more questions about your account? Check out Frequently Asked Questions or Troubleshooting.