Even if you're not close with your boss, chances are you have a working, first_name_basis relationship. Maintaining that in your official letter is okay. You don't need to keep a Victorian level of formality here__just make sure it's a professional tone.
A Professional Greeting. Your letter should be addressed to your boss. There's a good chance that this letter will merely end up in a file in HR, but it starts with your boss. Assuming this is someone with whom you work reasonably close with on a daily basis, it doesn't have to be super stiff and formal. Specific and professional is the right tone here.