Dear Jackson,
Hey... this is for your teacher , ok? .... So do you call your teacher by name? .... Even if you do, I guess for this task you better address him as Mr. :D
Don't begin with "I am really sorry to bother you." Your shooting yourself in the leg by beginning with that.
I too think that's not wise... Start with greeting him, and since he's your teacher, you need to show respect throughout this letter;
I hope this letter finds you well. I'm writing to you in hope of asking a favor. ... Now tell what you want him to do
Do you think I should make a simple self-introduction?
You don't need to introduce yourself as your teacher knows you already. But you need to give him the details that he requires to write about you.