I just received an e-mail from Apple saying a game purchase was made using my account

There are numerous scams like that. If you entered your Apple password, you should change it immediately. If you gave your credit card information, contact the credit card provider.

Also see: If you think your Apple ID has been compromised - Apple Support

If the only indication of the charges is from an email or text, be aware that it may be a phishing attempt to get you to log in to a site that appears to be Apple and divulge your password or credit card information. See: Avoid phishing emails, fake 'virus' alerts, phony support calls, and other scams - Apple Support

and Identify legitimate emails from the App Store or iTunes Store - Apple Support

If it is a real charge, see the following instructions. They include information on contacting Apple Support if the other steps don't help: If you see apple.com/bill, itunes.com/bill, or an unfamiliar charge on your statement - Apple Support

Apple will always address you by your name or the name they have on file for you, not Dear Customer, Dear Client or by using your e-mail address.  The e-mail will be from @apple.com or @iTunes.com. E-mail addresses can be spoofed. You can go to Mail/View/Message/Show all Headers to see more. Apple e-mails will never contain an attachment. Apple will never request personal information by email such as Social Security numbers, your Mother’s maiden name or full credit card numbers .

The only exception to the above I have noticed is if you order something from the Apple Store (apple.com), your receipt will be addressed to Dear Apple Customer. That is a receipt for a purchase you initiated.