Find out where your celebrity of choice is working or has recently worked. The International Movie Database is a great source of information on current and upcoming projects for nearly every actor in the business.
Look up the studio or production company addresses. Every television series has a production company and network attached, and every released movie has a studio of origin. Letters addressed to the celebrity in care of those entities will usually be forwarded to them.
Search out who the celebrity is represented by. A celebrity will have an agent, lawyer, manager or all three. These aren't unknown people and can be found through various means, such as published directories or online services.
Send a letter directly to a theater. If your celebrity is currently working on Broadway, you can send a letter to the actor directly to the theater.
Use fan clubs as another source for celebrity addresses. The more popular the celebrity, the greater the chance that he will have his own fan club. These can usually be found through a Google search and might help provide contact information.
Try looking up celebrities on their own webpage or social network sites such as Facebook and MySpace. For publicity purposes, celebrities will often have thier own websites in order to stay in touch with their fan base. You might be able to find a mailing address there.