Find the fan club, or contact address, for the famous person. Most celebrities have fan clubs that are listed on their official website. Other famous people, such as politicians, also have official websites. For example, to contact the President of the United States, go to WhiteHouse.gov and click on "Contact Us." The address for the White House is listed on the page.
Address the envelope to the contact address given for that famous person. Include your mailing address in the return section on the top-left corner of the envelope.
Write the letter. Include personal information about why you're a fan. Also request a reply. Enclose your letter in the envelope.
Include a picture, album cover or other item that you want the famous person to sign as well. This of course is not necessary, but if you desire an autograph, this is one method of getting one.
Include a self-addressed and stamped envelope with your letter and the signing item. Send the letter.