Doc was referring to the Hanna Barbera cartoon character from the 1960s, 70s and 80s called Huckleberry Hound. Doc was saying that he would be happy to behave like a wise-cracking, mild-mannered southern dog, if that's what was needed of him. Doc, though we was intending to be helpful, was very drunk, as usual, and was saying very strange things. Doc Holiday, of course, could only have been seeing into the future because Huckleberry Hound would not have been created for another 75 years. That's amazing, isn't it?
When Doc Holiday says "I'll be your Huckleberry" in 'Tombstone", he was saying 'I'm your guy'. It basically means that he's the right man for the job and that he will get what needs to be done done correctly.