Hi, *belle*, I wouldn't worry too much about sounding cheesy, I'm sure they'd appreciate it anyway. I think it comes easier when writing to someone you particularly like.
I think it depends a bit how old you are, how old they are, and exactly what you think of them, whether they're your favourite character, or someone you could imagine as a friend, or if you think they're good looking (/beautiful for the ladies)! Tell them why you like their character - or even, politely, what you don't like so much about their character. Maybe tell them what storylines you've enjoyed, what you might like their character to do in the future, tell them a little about yourself, you can ask questions, but it's not too likely they have time to respond in detail. And probably ask for a fan card at some point so they're in no doubt that you'd like one!
I wrote my first letter to Nic a few weeks ago, so I don't claim to be an expert. This is basically what I did, and just made it pretty clear how much I like him. (I hope he didn't think any of it was too cheeky, given that his partner recently had their first child!) I just really wanted to say thanks to him for all his work on the show, that he's most definitely appreciated. I found it difficult to decide how much to write, I was worried about writing too much, but I tried hard to keep it concise. I think longer is better because it shows you're more interested, but there's no rules, and if it's shorter, it takes up less of their time to read.
Just some thoughts, good luck!