To say that your friendship progressed very quickly, you could say
I felt we were old friends from childhood.
I felt we got to know each other very quickly.
In the short amount of time we knew each other, we just seemed to click.
You were really easy to get to know.
If you say
I felt like I was really close to you
I feel like I'm really close to you
it may be interpreted as having romantic overtones, more idiomatic would be
I felt we were really close friends.
Of course you may want to end with something like
I hope we will continue to be friends in the future.