How does a step child go about taking on the last name of a step father?

If you'll pardon the pun, you've already taken the first step by being accepted by your step-father as his child. With that record on file at your area courthouse, go to the courthouse and locate the section where you can change your name legally. And congrats. Adoption, the biological father has to sign a paper of name change and rights.(sometimes all rights sometimes whatever you want)