I'm only counting on-screen kills, and only when zombies kill humans or humans kill zombies.
Warning: If you haven't watched this episode yet... spoilers ahead!
Episode 1: "Days Gone Bye"
Morgan kills 3 -- we first meet him walking up to a zombie in the street and shooting it in the head, gangsta style; later on, he picks off two with his rifle. Rick kills 10, starting with the little girl in the cold open. Do you think this is the first time in television history that the main character has shot a little girl in the head, 4 minutes into the first episode? I'd have to guess yes. But of course, the little girl is a flash-forward; he actually kills his first zombie with his Joe DiMaggio impression outside the house. He also shoots the idiot deputy, the lovely crawler in the park pictured above, five while under the tank, and finally the soldier in the tank.
Oh, and also: 12 shots fired, not a single wasted bullet.
Humans Killed: 0, Zombies Killed: 13