I was adopted and I don't want that adopted families name any more I want my birth mother to adopt me back how do I start the process and how much would it cost me?

How old are you?

Do you know your birth mother?

And unless you are a adult, there is nothing you can do,

You don't start the process. Your birth mother - who is no longer LEGALLY related to you - can either move to set the adoption aside (and will almost undoubtedly lose that argument) OR file a Petition to adopt you. She cannot do so without the written consent of both of your LEGAL parents (who are your adopted parents).

If it's just about a NAME, and you are an ADULT---then go down to the courthouse and change it already.

If you are NOT an adult, you need to be having these conversations with your adoptive parents.

The biggest question here is how old YOU are.