It's conventional to drop the subject "I" in informal speech or writing in this kind of expression.

By the way, American isn't a language!

I would just remove the word "very".  "I hope you're doing well" or "I hope this note finds you well" are more formal with the subject "I".  "Hope you're doing well" is less formal/more casual.

I think you should add a subjecti to the sentence. " I " hope you are doing well.