Start Duel Mode and play until you get to the 3rd duel which is against Old Man Clanton.
Follow his steps (I think he always walks to the far right first and then back to the middle)
and be sure to keep your focus and speed high as possible. As soon as you hear the heartbeats you can draw your weapon and first shoot the vulture over his right shoulder and then him. You will kill him dishonourably but still obtain the trophy.
After a couple of tries you should be able kill the vulture and him as I described above.