Linking an offline Xbox 360 avatar to an existing account.

Connecting to Xbox live with an existing account is exactly what i am trying to do.

You cannot link one account to another. Sorry to say it's just not possible. Unless you are trying to make your offline account an online one, then this should help (I got it from the Xbox support page)

To convert your offline profile into a gamertag, make sure that you have a Microsoft account available that's not currently associated with another gamertag.

1. Sign in to the profile that you want to convert from an offline profile to a gamertag.
2. Press the Guide button to bring up the mini-guide.
3. Go to Settings and then select Profile.
4. Select Join Xbox Live.
5. Enter your first name.
6. Enter your Microsoft account email address.
7. Enter your Microsoft account password.
8. Verify the information on the screen.
9. Verify the current privacy settings.
10. Review the
    Microsoft Services Agreement and then select I accept.
    

Here is the link if you want it
https://beta.support.xbox.com/help/account-profile/profile/change-xbox-live-gamertag

Hi you can restore your gamertag from the cloud via the XBOX 360. Does your console go online and connect to XBOX Live?

Sorry but you can't merge 2 accounts.

@StrangeSubset1 Hey , maybe this can help you out : From what I understood your boy has a offline account on which he played only offline until now. Initially when he made this account I guess he made it as a child account and he added his real age there
( less than 18 years of age) . This means that now your boy's account can join live and play on xbox live only if it has a parent account linked to it. if your son's account is not like to a full adult xbox account( parent account) he wont be able to play
on xbox live because somebody needs to accept the terms of use and mange the content controls of this child account. What I would recommend you to try is to add your sons account to a parent account. You can try to add this child account to a New Microsoft
account ( it would be best to creat a new one for yourself) . I would recommend to contact Xbox Customer Support explain to the your situation and they can guide you on how this can be done , and what can be done in your situation. A resolution will be found
with a bit of luck and patience Good luck