You will need:
A Phone
Soft, relaxed, and somewhat friendly tone of voice.

1. Sign into your account on [ Register or Signin to view external links. ] , go to the 'Accounts' tab and go to 'Upgrade billing information'

2. What you want to do is change your Surname to something similar to your Gamertag for example; GT: 'Quickscope' (or something similar) say your name is 'Bob Quick'.

3. Now, call (800) 4MY-XBOX or (800) 469-9269 (They're the same number.)

4. Depending on which country you're from, follow the prompts from the automated Microsoft voice until you get a customer support member.

5. If they don't ask for your name right away, then instead ask what your problem is, you would say something like this:

"Hey, just the other day I was reading the Code of Conduct and saw that you shouldn't have personal identifying information in your Gamertag, currently my Gamertag is 'Quickscope' and it is very similar if not the same to my real name 'Bob Quick'".

6. The consultant will run you through some stuff just listen and follow what he/she says.

7. If they are telling you that you need Microsoft Points you can either stay on the line and tell them a story as to why you do not have any money to buy Microsoft Points or just hang up and try again.

8. They will quickly brief you on why the information shouldn't be in your Gamertag, if they are hesitant to change your Gamertag for free either; tell them that you've read their Code of Conduct and hint to them that they are liable for the problem and that they should change it. (They don't tell you not to put your last name in your Gamertag) Ask them if you can talk to their supervisor.

9. If you have done this correctly, around 5-10 minutes on the phone, they will tell you that the next time you go on your Xbox 360 console or on [ Register or Signin to view external links. ] you will be able to change your Gamertag, due to them allowing it.

This should work 95% of the time depending on your Social Engineering skills (SE'ing skills), as for me it worked first try with no problems. (It has worked for me, as well.)

Recommendations / Tips:
# If your Gamertag is highly unlikely to be your real name try using an accent, for example 'Sniper', say your name is 'George Snipler' in some European accent, or wherever else you'd like it to be from.
# Be nice and understanding but at the same time forthwrite.
# You must know all the information for your profile (Credit Card information, Secret Questions, Previous Passwords, etc.)
# Be persistent if you can't get it to work (eventually it will, as I said depending on your SE'ing skills), don't give up.
# If you're having problems thinking of a name, try [ Register or Signin to view external links. ] and search for something similar.
# Try to remember their name so you can seem polite and also if you call again you can hang up if it's them!

This will work even if your Gamertag is unlikely to be your name, just use some common sense and make it sound believable.