[rewboss]

Generally, you only write things like "gonna" or "goin'" if you have some reason for wanting to record somebody's non-standard speech patterns. For example:

He opened his mouth and said, "Listen, buddy, I'm gonna bust your ass!" That's when I realised he was not only angry, but American.

Otherwise, we don't write like that. We do write established contractions like "shouldn't" or "they've", but otherwise, even in very informal contexts, people tend to write "going to", not "gonna".

If you browse some English-language message boards, you'll find that "gonna", "wanna" and similar expressions almost never appear, except perhaps for humorous effect. (You will see lots of typing errors, but they're not deliberate.) If you use these expressions in a personal letter, you will probably appear insincere or uneducated -- even if the person you are writing to actually speaks like that.

Another point: Be careful when trying to use slang, either written or spoken. If you get it exactly right, people will be impressed; but if you get it even a tiny little bit wrong, people will just laugh at you.