What are you writing is an epistolary novel, though you probably won't take it to the extent of an actual novel. I think the Diary of Anne Frank is a good example of this, since it tells a story through a diary. This isn't exactly a letter, but I don't think think they are that different. Here is a blog of quite a few excerpts, and hopefully this can give you a general idea, the difference being that the letters are directed at someone and Anne Frank's diary is moreover her thoughts. Fundamentally, they're nearly the same.
Obviously, you'd write in the first person and just describe things. Think of it as rambling of sorts. Describe an event, gossip, etc. I always thought that it would be interesting if two people wrote from two different sides of a war. That way, readers could get both sides of the war and a mixture of events and opinions. Just some food for thought. You don't have to do it that way.
Edit: If you're interested, there is an anime that you can watch that shows two characters contacting each other through text messages. The anime itself is only one episode, so it's technically a short movie. It's about 25 minutes long and it's called Voices of a Distant Star. I just remembered it and I thought it could be a good example of sorts. Don't worry, it's a really good anime, but since it's an anime, it might be harder to get information and style from it.
Another thing is that you're writing through two (or more?) character's 'voices' so to speak - ie you can play around with some emotion and the person's mannerisms (how they write or word things, etc) in telling the story and building up their character too.
Anyways, I actually read such a novel before presented in this way. It's Letters From the Inside by John Marsden (guy who did the Tomorrow Series if that rings a bell?), and although this isn't in letter form it's a journal form and a darn good read too from memory by him: So Much to Tell You. If you have time and have access to them I'd think they would be of help.
Another good example, believe it or not, is Dracula. I had to study it in Year 10, and I was most surprised to find that it was entirely made up of letters, diary entries, scribbled notes and newspaper clippings. It was a fascinating way to present the narrative, and although I personally found the book as a whole boring as sin, it's definitely worth a look if you want a classic example of the genre. Stoker does a really good job of using the format to tell the story.