In order to have a profile picture in Kings of Pool, you'll need to connect your Facebook to your KoP account from the KoP splash screen. Your FB profile picture is then automatically used for your KoP picture.
This is currently the only way of getting a profile picture. If you aren't connected to Facebook, you're considered a "Guest Account," and your profile will have a grey cartoon silhouette as a profile picture.
If you wish to change your profile picture in Kings of Pool, you just need to change the one on Facebook.