How can I go of adding my last name to my daughters birthcertificate?

Why isn't it there in the first place?

You can legally change a child's name for free up until the child reaches 1 year old - changing the name/adding another last name, etc. I take it, you'd like to hyphenate her last name to show your and her father's last name?

Go to the courthouse (where you would go for a copy of the birth certificate) and ask for the form to change the child's name - fill it out and go from there!

Good luck!