In order to set a profile picture, you need to have your Trivia Crack account linked to your Facebook account. This way, since Facebook makes sure that the pictures comply with the rules, we prevent users from uploading inappropriate pictures.
If you don't link your account to your Facebook account, 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 account but you don't want your information to be displayed, you can hide your picture and/or name by going to Menu>Settings>Account>Facebook.