I recommend that you check StarArchive for the results of other people's attempts at writing to the addresses contained on this site, as well as slightly more up-to-date information.
It is often better to write to a venue just prior to the celebrity performing in an event (such as a concert or play). Such a letter should be sent and timed to arrive 2-3 days prior to the event. The success rate for such letters far exceeds writing to standardised addresses. A good place to find out about events is TicketMaster.
There is nothing worse for a star than receiving a long letter. Remember they receive bags and bags of fan mail daily, so keep your letter short!
For a quick response, enclose a stamped envelope addressed back to yourself. If you are writing to a star who lives overseas, enclose an international reply paid coupon, which you can get from your post office - your thoughtfulness will be much appreciated by the star.
Always keep your letter informal, as if you were speaking directly to them.
If your handwriting is not up to par, print or, better still, type your letter. Make sure your letter is easy to read and you will be much more likely to get an answer.
Make sure to include your name and, of course, your address in your letter.
Sending a birthday card to your favourite star is a wonderful way to show you care, and stars love receiving them from fans.
For the sake of privacy and safety, stars generally do not have their home addresses and phone numbers available. In reality the best place you can write to a star is care of their agent or official fan club. These are the two most certain ways to get your letter to them, as they are in daily contact with both their agent and their publicist. Letters to the studio where they shoot can go missing, especially if their show has finished shooting for that season.
Keep in mind that I can't guarantee that you will always get a reply, or get what you asked for. Sometimes celebrities change their agents; this doesn't happen often but if it does you are at the mercy of the previous agent to either forward your mail on or return it to you.
Remember: one person who writes to another makes more impact than thousands who are silent! If you don't write to your favorite celebrity you will never get a reply!

Thanks to David Hartnell