Instructions
Turn on the Xbox 360 and go to the dashboard. If a game loads instead of the dashboard, press the "Guide" button in the middle of the controller and then press "Y" to access the dashboard.
Scroll up to the "My Xbox" menu on the dashboard and then scroll right to "System Settings" and press the "A" button.
Scroll down to "Memory" and press the "A" button, then press "A" again to view "All Devices."
Scroll down to "Gamer Profiles" and press "A."
Scroll to the gamertag you want to delete and press "A."
Scroll down to "Delete" and press "A" to delete the gamertag, then select "Delete Profile Only."