I'm going to add a bit more advice to the excellent post by Groo:

He already covered limiting your range to that in which you can keep all your shots on a paper plate. That is a Golden Rule of hunting deer with a pistol.

The second most important rule, in my mind is this: Follow up every shot, whether you think you hit OR missed. Do NOT expect the deer to fall when you shoot it. For every one that drops in his tracks, probably ten do not. Some won't show any sign whatsoever of being hit. Go to the spot where they were when you shot and look for any evidence you can find of a hit. This is where full penetration really helps--exit wounds almost always leave blood on the ground, entrance wounds do not. Practice up on your tracking if you're going to hunt with a handgun, especially a .357. It doesn't have much margin of error.

Profit from others' mistakes and advice. Don't learn this the hard way, as I have.