The Old West wasn't wild just because of the people. Even the animals wanted a piece of the action. Hunt animals that come across your path, and maybe everyone will get dinner tonight. These missions are triggered by talking to the Indian Hunter near Piper Lake.
You begin to hunt by talking to the native American hunter. First you need a bow, as apparently killing them with a bullet takes no skill and the bow is a fantastic weapon. Next you will need to find your prey going only on the hunches the hunter will give you. Once you close in on the area of your prey, you will get a message saying that it is near. That's your cue to get off your horse. Now you creep up on your prey by crouching and treading lightly, then aim with the bow and zoom. You will get another message telling you when you're in range. Aim for the head and let arrows fly. If you miss or hit the body, the animal will generally run off and you have to find it again, but sometimes it'll just run around in a circle leaving you to aim again and this time don't miss. Remember, aim for the head, because no matter how many arrows you put in the rest of the creature, it won't die.
The Grey Wolf can be found in one of two places – either by the bridge to Dodge, or by the ranch where you chase Chavez y Chavez. Either way, the strategy is clear – sneak up on the wolf and shoot it in the head with the bow. Using the bow is critical in these missions, as if you use your guns, the result is that you destroy the pelts and lose money for it; for the same reason, only use the Apache or the Blackfoot Fire bows, not the Dynamite Bow. When outside Dodge city, the best place to be is on the other side of the rail tracks away from the wolf, because if you miss, the wolf will run away in your direction giving you another chance to hit it in the head.
Great White Buffalo
It's on the plains just outside the ranch. The best position for you be in is by the hilly side of the plains, as if you miss, it'll run towards you giving you another chance to hit it right between the eyes.
This she-wolf hangs around either Piper Lake or the Apache tribal grounds (where you meet Honest Tom after the steamboat ordeal). If you're around the Apache's tribal site, then approach from under the bridge, as once again this is the way it will run towards you if you miss or spook it, giving you a chance to hit it in the head.
Great Mountain Lion
This appears to the east of the hunter in Devil's Canyon. Go past the rail tracks and into the track that has brown soil. Now this is an odd one, as it will just get running around in circles if you miss, so the only danger is if you run out of arrows.
The Black wolf appears in between The Ranch and Empire (Where you free'd cattle, and sometimes Bandit Attacks) Appears on the left of the entrance. It will try to run out of the area, through the entrance/exit. You can also find it near the mineshaft in the plains.
Hunt The Grizzly
This is a mission, so you don't have to use the bow, thankfully. Head over to the Empire. You're guided now, so it shouldn't be hard to find, but in case you can't find him, he's at the end of the gully that runs alongside Empire. The first thing to do once you've seen it, is get off your horse and get it to a safe distance away, but not too far, as you'll need it later. Fight the grizzly as you fought the bear back at the start of the game; avoid its charges and blast it with the shotgun. After it has taken a large amount of damage, it'll run off. (If you can hit the grizzly a few more times before it runs off, you can skip the entire second battle.) If you put your horse somewhere safe, you can follow the bear, otherwise it may get too far ahead and the mission will be failed. Once you've cornered it, start the battle again. Be wary, as the grizzly will have full health again. Once you've done this one, you've finished all the hunting missions.