Rule #1: Don't type your letter.
Rule #2: See Rule #1!!!
I have had success sending letters with my own return address label, looks more like personal correspondence.
Check interviews with various celebrities, or "where are they now" type articles to find where someone is working. A lot of times if someone gets a request at work or via venue, they are more likely to sign, and you will be more likely to get a real signature rather than sending via agency for movie stars or via team for active players.
Home addresses work ok too, some people will refuse stuff sent to their homes though.
My Dad always used to joke with me saying "Dear _____, you are my favorite ____...." and sending these out in volume, that is about what it gets to be like. I collected TTM from about 1992 on, and just got out of it in 2016. So in 24 years, I have had a lot come back. I have had a lot of people comment on my letter, "Thanks for your kind words" etc. and just as many come back signed with no comments, but I think a sincere, handwritten letter takes you far.
One note: Shirley Temple got a birthday card from us, and she signed a small photo I had enclosed, personalized to my wife and myself. HUGE success, my wife loves her movies (may she rest in peace). I wouldnt overuse the birthday card with photo and SASE, but thought it wouldnt hurt, especially since she was aging and signing less consistently at the time we sent it. Of course this was probably 2008 or thereabouts.
Hope that helps, its a lot of fun to get home and check the mailbox! It was even more fun for me to sell 99% of what I collected over time and go to Europe with my wife twice, so I would say the hobby has paid off!