Okay, I found this on another site -- this guy found some good references showing that it's the dog being hunted, not doing the hunting, but the origin remains hazy --
"Wait! Don't accept that half-assed Googling as research. Here are some true citations. Also, smackfu's reading is wrong: the person being hunted is treated as a dog. The hunter is not hunting like a dog. Think of a rabid animal, perhaps, or one that has taken to killing chickens as a habit, and so must be killed. Dogs, though much loved as pets by some, have often been (and still are) treated as nuisances, low creatures, and something not worth attaching a lot of sentimentality to. I've included some citations for "like the dog you are" to illustrate this point.
23 May 1878 Indiana (Pa.) Progress "A Close Shave", p. 2: "A man will come after me who will hunt you down like the cowardly dogs you are. He will never rest until you are driven out of the country."
28 Feb. 1896 Newark (Ohio) Daily Advocate "A Creole Romance" p. 3: "It is you who who have disgraced her and ruined their home. Old Dominique Farge will kill you. He has sworn to hunt you down like a dog."
4 Apr. 1905 Elyria (Ohio) Evening Telegram "Woman Threatened To Kill N.Y. Lawyer Is In Charge In Letter" p. 4: "I tell you with malice and premeditation that unless you do, I will come to New York and disgrace you publicly first, then shoot you like the dog you are."
5 Apr. 1929 (Lincoln, Neb.) Evening State Journal "Beau Ideal" p. 14: "You [...] have taken French gold and would use it to bribe a servant of France [...] if I live, I will command the firing party that shall shoot you like the dog you are."
4 Aug. 1929 Los Angeles Times "The Wandering Gentile" p. F8: "Try any of that sort of vileness in my district, Savaran, and, in spite of all I owe you, I'll hunt you down like a dog."
posted by Mo Nickels at 10:01 AM on February 4, 2005"
Hope it helps.