The only correct one is "A good friend of mine". The second one, "A my good friend", makes no sense. You could say, "My good friend", and it would mean the same thing. Here are some examples of when you could use the different sentences:
"Andrei is a good friend of mine" or "I'd like you to meet my good friend, Natalia."
They both meant the same thing, it just depends how you structure your sentence in which one you use.