In order to set a profile picture, your account must be linked to a Facebook profile. This way, since Facebook is in charge of making sure that the photos comply with the safety rules, we avoid the possibility of inappropriate images being used.
If your account is not linked to a Facebook profile, a random avatar will be assigned to you, and you will not be able to change it.
If you want to link your account to a Facebook profile, go to the game's Settings by tapping on the gear icon in the upper right-hand corner of your screen and select "Link with Facebook".