Instructions
Find out your friend's Xbox Live gamertag. This is a personal identifier that each player with an Xbox Live account creates.
Turn on your Xbox 360 and log in to your game profile.
Scroll over until the "Friends" tab at the top of the screen is highlighted, then press "A."
Select the "Add Friend" option and then select "Search."
Enter in your friend's gamertag and then press "Done."
Select "Send Friend Request" if the gamertag was entered correctly to send your friend a request to be added to each others' in-game friends list.