Wait until it says "Draw". Immediately press and hold the left trigger. This will begin dead eye mode automatically. Your reticule will appear below the other gunman's feet. Use the right stick to move the reticule up to the area of the body you wish to shoot.
You should notice that the reticule strobes, the pieces move from far away from the center of the reticule to near the center of it; the closer to the center, the better!
When the aiming reticule is over the part of the body you wish to shoot wait until its pieces are very close to the center (of the reticule area) and then press the right trigger to paint the target.
The closer they are to the center, the higher your score. The higher your score, the larger an area is filled in the vertical bar to the right. If the other gunman's first line in the bar is higher than yours, you will lose. So, click when the reticule pieces are as close as possible.