"Hi!" is a sentence by itself.
Dear Jane,
Hi! I bet you thought you'd never hear from me again. I've been very busy since the move but I hate to lose touch with my friends, so I thought I'd write you.
The correct way to use "hi" with a name is the same as "hello"- with a comma separating the two, and then end punctuation.
"Hello, Jane. It's good to see you back at school."
"Hi, Jane! Wow, you look great!"
In these days of text messaging and other truncated communication, one does see things like
"hi Jane - sorry to ask this at the last minute but could you put those tickets at the front desk & I'll come by for them later, thanks Bob"
This is comprehensible, but not strictly grammatically correct. That's not to say I never write emails like this myself, however :)