This article explains how to log into your Game Center account using a Mac. When you sign up for an iCloud account, a Game Center profile is automatically created as well. You can only log into one account per device.
Steps
Step 1. Click on the Apple menu
The icon looks like an apple and is located in the upper left corner of the screen. A drop-down menu will open.
Step 2. Click on System Preferences
It is the second option and is found under "About This Mac".
Step 3. Click on Internet Accounts
The icon is represented by a blue circle with a white "@" inside.
Step 4. Click on +
Click on the "+" button at the bottom left of the account window.
Step 5. Scroll down and click Add another account
It's in the box on the right, at the bottom of the list of accounts you can add.
Step 6. Click on Game Center Account
It is located next to the icon that depicts circles of different colors and sizes.
Step 7. Enter your Apple ID and password
The Apple ID is usually created the first time you set up a Mac or iPhone.
If you don't have an Apple ID, click on "iCloud" in the "Internet Accounts" section. Then, click on "Create Apple ID"
Step 8. Click Login
This will log you into your Game Center account.