I always suggest opening with a compliment — the “kill ’em with kindness” approach — that way you don’t just sound like you want something from them but you’re here to contribute something to their obviously stellar output. Plus, everyone likes a little ego stroke, especially in Hollywood.
Open the letter by congratulating them on the success of their latest film and tell them how much you love one of their smaller, critically-acclaimed films. They will appreciate you noticing one of their lesser-known, more artistic efforts. I also suggest doing some research and pointing out an obscure fact about them and their work that could only have been known by someone who took the time to do their homework.