Write no more than two paragraphs and make sure that they appear on one page. (Nobody has time to read through pages of correspondence.)
In the first paragraph tell them why you like them. What you've seen of their work that you're particularly fond of and congratulate them on their current work.
In the second paragraph tell them that you would really like their autograph on a picture.
If possible SEND THEM THE PICTURE THAT YOU WOULD LIKE SIGNED! Photos are expensive - particularly for performers who aren't mega-stars with tons of work (ie MONEY). ALWAYS include a SASE for them to return the picture. By enclosing both of those items, you aren't costing them any more than a couple minutes of their time. Something a real fan should be happy to do!
Be aware that you may not hear anything from them. Or you may hear - but it will come much later than you think. I sent a photo to David Warner when he was appearing in Major Barbara. I told him how much I enjoyed the show and that I didn't want to bother him at the door, but I really would like his autograph on my picture. I sent the photo to him, in care of the theatre, very early in the run. The show completed the limited run (about a month after I sent the photo) and then ANOTHER 2 months passed. I completely wiped it out of my mind. Suddenly there it was on my doorstep! Not only autographed, but with a personal message written on it.
I also sent a picture to Liam Neeson when The Crucible was playing. About 2 weeks later I got my SASE back. The photo I sent was in there - unsigned. But there was also an autographed Playbill in there from the show. So - I was happy! I sent a Playbill to Frank Gorshin when "Say Goodnight Gracie" was on. I got it back in 4 days with a lovely personal message written on it.
I also sent a photo to Bill Pullman at "The Goat" and never got it back at all. I figured he was just not someone who responded to his fans and wrote it off. Later on I actually met him after the show one evening and he couldn't have been nicer. He spent a good 20 minutes chatting with the folks who were there, autographing Playbills, and posing for pictures. So you just never know!
Good luck!
T'was the night before Christmas, and all through the house
not a creature was stirring....just me and my mouse.