Kill 2 Cougars with your knife and collect 2 Cougar pelts.
Like the boar, the cougar will generally be on top of you before you even know it's there. The difference is that the cougar is both faster and more powerful. Two hits from this bad boy will generally be enough to leave you dead in the dirt, so keep your eyes open and your knife ready. Aside from that, cougars are just another animal that’s more rare.
Collect 5 Raccoon pelts, 5 Fox pelts and 5 Skunk pelts.
After taking down boars and cougars, you’d think killing animals of the small, furry variety would be a welcome change. It’s not. While these guys lack the raw killing power of your previous prey, they are hard as hell to find and you need a lot of them. The key here is persistence. Keep it up and you will eventually find what you need.
You might try hunting at night, as the nocturnal nature of these animals has them out more often when the sun has set. Also, don’t use too powerful a weapon when shooting them. While it may have been a glitch, occasionally when using the shotgun and buffalo rifle, some smaller animals seem to just plain explode, leaving them quite unharvestable.
Collect 5 Elk meat and 5 Bighorn skins, not counting those already in your possession.
The difficulty here is again that elk and bighorn are rare and generally located in a single area. This means that you’re in for a long hunt. Just be persistent and eventually you’ll accumulate the goods you need.
Kill a Bear with your melee knife and collect a Bear pelt.
On the one hand, you only need to hunt one bear to complete this mission. On the other hand, you’re restricted to hunting a freaking bear with just a knife. Bears are vicious, brutal animals and chances are that the first time you encounter one it’s going to tear you apart. Keep at it and eventually you should catch a break. A bit of advice - you can use the dead aim to slow down time for the knife as well as weapons, so try it out if you’re having difficulties.