Instructions
Turn on your Xbox 360 console and sign in to your Xbox Live account. Press the "A" button to select your gamertag from the "Profiles" screen on the dashboard.
Use the left joystick to scroll down to "Friends" on the left-hand side of the screen. Press the "A" button when the option is highlighted to select it.
Highlight "Add Friend" by moving the left joystick, and press "A" to select to the option.
Input the gamertag you're searching for by using the left joystick to highlight the letters or numbers in your friend's gamertag, pressing "A" to select each letter.
Press the "Start" button, which is labeled as "Done" on the screen, once you've entered the complete gamertag.
Select "Send Request" by highlighting the option and pressing the "A" button on the next screen once Xbox Live has searched for and found the gamertag you entered if you want to add the person as a friend. You can type a personal message to send with your request if you choose.