My daughters father and I got married last August but when she was born I had given her my last name, not his. Now that we are married I am changing my last name to his and I want to change her last name as well. Do I have to go through the courts to do this?

You are saying that she was born before you married your husband?
Then he has to adopt her formally. Start on the phone to your local or county courthouse.

First, any question on law needs to include your general locale as laws vary by area. How long after she was born did you marry? Some areas will award legal paternity if you married within a year of her birth. Otherwise Joy is right that he may have to adopt or prove paternity.

Is she his biological daughter?

Is his name on the birth certificate?