I have been divorced for almost 4 years and have a 5 year old son from that relationship. I am going to be getting married in 1 1/2 years. We also have a son together. My fiancé has been with me since my older son was 10 months. Needless to say he calls him daddy and his biological father by his first name. I would like to add my fiance's last name to my sons last name once we get married. For an example Adam Adams-Johnson. Do I need to get his biological father's approval or is there a way for me to do that without him?

Yes you are the legal parent therefore you may change his name. Here is a link i wanted you to read hope it helps.
Re: Changing my children's last name.
Also this one gives you numbers to your state
Changing Birth Certificate

To LEGALLY cahnge the name you have to go through a Family Court judge. If the bio father objects the judge may not grant it.

However, you can use any name you want as long as there is no intent to defraud.

First, please do not use PMs for followups. Just reply in the thread.

But yes, when you register your child you can inform that of what name you want her to use. You will have to inform them of her legal name, but there should be no problem using a hyphenated name.

Thank you for the advice, that answers my question.