You can change your picture by changing your Facebook account picture! To change your Facebook account picture, you can follow the steps here.
If you have changed your Facebook account picture but your game picture has not updated, navigate to your settings page in Snoopy Pop. Once you've made sure that you are currently connected to Facebook, close the app completely and reopen it. When your app has reloaded, your picture should be updated.
If you are not connected to Facebook, you will have to sync your game to a Facebook account to be able to change the picture!