If using iOS 7, you should look in your Settings app for iCloud >> Document & Data >> Alphabear. Make sure that it's enabled at all of these levels.
If using iOS 8 to iOS 10, you should look in your Settings app for iCloud >> iCloud Drive >> Alphabear. Make sure that it's enabled at all of these levels. You might have to upgrade your iCloud to iCloud Drive if you have never done so before.
If using iOS 11 or above, open your Settings app, tap your account name, then tap iCloud, then scroll down to the section that says iCloud Drive and make sure it's enabled. After you enable it, make sure Alphabear is listed underneath it and that it is also enabled.
You must have a Game Center account and be signed into it. You can sign into your Game Center account by going to your Settings app and looking for Game Center. Enter your Apple ID and Password, then click the button that says Sign In.
Make sure that you've done everything listed in the Requirements section above.
Launch the Alphabear app.
Your Alphabear app should greet you with at "Welcome back, [your name]" if you're already using Alphabear with Game Center. If not, open Alphabear's Options menu and click the Sign In button.
If you have Alphabear version 01.09.02 or higher, play one game. Then skip to the last step--you are done!
If you have Alphabear version 01.09.01 or lower:
Now that you've signed in, play until a screen comes up asking you if you want to turn on iCloud saves. This occurs every 5 games (unless you at some point in the past saw this screen and clicked the middle option to say that you never want to do this).
Click the first option on that screen.
Done! This particular device should now be saving its Alphabear progress to the cloud.