Only Alliance Leaders can change the Alliance emblem.
To update your Alliance emblem:
Go to the game's Home screen.
Tap ALLIANCE.
Tap the current Alliance emblem in the corner of the screen.
Select Edit.
Tap the emblem icon to view the available emblems.
Choose the emblem you'd like to use.
Tap Select.
Tap Confirm.
Your new Alliance emblem should appear immediately after the change is confirmed.