Thanks for the question. I understand you are receiving a prompt that you have authorized your maximum number of computers. If you wish to quickly reauthorize all computers, and free up the option to authorize this one, follow these steps:
To deauthorize all computers associated with your Apple ID
If you need to authorize your new computer and are unable due to already having five authorized computers, you can deauthorize all computers by doing the following:
1. Click iTunes Store on the left side of iTunes.
2. If you're not signed in to the store, click the Account button, then enter your account name and password.
3. Click the Account button again (your Apple ID appears on the button), enter your password, and then click View Account.
4. In the Account Information window, click Deauthorize All.
Note: You may only use this feature once per year. The Deauthorize All button will not appear if you have fewer than two authorized computers. If you need assistance on using this feature, please contact iTunes Store support via email (http://www.apple.com/support/itunes/store/).