What is useful when writing letters as other characters is using dialect. In some books, dialect is reflected in the way certain words are spelled. They won't be the correct spelling, but rather a phonetic spelling. Some modern day examples would be:
Gonna (Going to)
Wanna (Want to)
Shoulda (Should have)
Obviously, these words are not spelled the same way, but you get a feel of how someone is talking. There are examples of this in classic literature or (if you listen carefully) you might be able to pick up on dialect while watching period dramas and can formulate how to spell the words just by sound.
An example of dialect in classic literature is in Wuthering Heights*. There is a character named Joseph that is a servant (not sure if he's in any of the movies, but he is in the book) who talks with a very thick dialect. Here is a sample of his dialogue:
"...Shame on ye! sit ye down, ill childer! there’s good books eneugh if ye’ll read ’em: sit ye down, and think o’ yer sowls!" (Chapter 3)
Because he is talking in a certain dialect that does not sound like, for instance, the Earnshaws that he works for, he would not sound like, "Shame on you! Sit down, ill children! There are good books enough for you if you will read them: sit down and think of your souls!" Writing it this way would not sound like the Joseph that is in Wuthering Heights.
It might help to form what you want to say in proper grammar and then add/replace dialect into your original sentence. For a sentence like:
"There are plenty of great books."
As Joseph, you might say:
"There's eneugh good books for ye to read."
There is more than one way to do this. Try experimenting with sentences and see what you come up with.
On a similar subject: grammar. Some characters have great grammar while others do not. Take for example Lucy Steele (Sense and Sensibility). Lucy is - shall we say it? - not the most educated character in literature. Therefore, in her dialogue there are some grammatical errors:
Right away, you notice that there was a grammatical error: it should be "I felt almost as if you were an old acquaintance". If you include these grammatical errors in your letter, they can really help your letter sound like it is coming from the character you are writing as. Be careful with this though. There are characters that use a lot of improper grammar, but there are also characters that only use enough to give the impression that they are not educated, but their bad grammar is not blatant. Take a look at your character's dialogue and see how their grammar is.
*Note: I am not fond of the book Wuthering Heights, but for our purposes here, it does a good job to show how to use dialect.
How helpful was this tip? If you have any questions, feel free to leave a comment. I'll try my best to help out.