Kill and skin 2 Cougars. But you must kill them using your Melee Knife.
A good location for hunting Cougars is: Near Gaptooth Ridge, a little north-east of it, but still below the Stead when you look at the map.
Another good place to find cougars is out behind Fort Mercer and towards Del Lobo Rock. In the early morning, you can kill six in about 5 minutes, just make sure you have lots of medicine!
Protip for killing with your Melee Knife:
1. Try locking onto an animal with your melee knife, wait for it to charge you, and then right before it jumps you do your first swing with your knife.
2. An alternate way to kill a cougar is by using a low power pistol to shoot the hind quarters/legs of the animal, and then switch to your melee knife for an easier kill.
Kill and skin 5 Raccoons, 5 Skunks and 5 Foxes.
A good location for hunting Raccoons and Foxes is: During the night in the Tall Trees area. And behind the general store in Armadillo, they will repeatedly spawn there if you circle around it several times.
A good location for hunting Skunks is: During the night in Rio Bravo.