Print off form DS-5504 (see Resources) if you just received your passport less than one year ago. However, if your passport is more than one year old you need to print off form DS-82 (see Resources). You can also obtain these forms from any U.S. Post Office.
Obtain a certified copy of your marriage license or whatever government issued paperwork you receive notifying your name change. You can obtain these from the country clerk office where the paperwork was originally filed.
Make a check or certified money order out to "Department of State" for $75 (if the current passport is more than a year old). If your passport is less than a year old you don't need to pay any processing fee.
Have two passport photographs taken of you. Your local post office can do this, and it is recommended that you have your's taken there because post offices know passport photo regulations.
Take all of the compiled information and take it into your post office. Mail out the information to the State Department address listed on the bottom of either form DS-5504 or form DS-82. The State Department highly recommends that you to send it via a traceable delivery method so you know when the proper authorities receive your paperwork.