Once you've either purchased or entered a code for Call of Duty Elite, it is linked to the Xbox gamertag that you are using. If you were to create a brand new gamertag, there is no way that the Call of Duty Elite service can be transferred over
Alternatively, If you just want to change your Xbox gamertag name, you can do so and continue to have the same Call of Duty Elite benefits on the account.
Also, If you make a new account and you purchase Call of Duty Elite again, you will not receive the founder benefits as they were a limited time offering.
Let us know if you have any other questions.