Hunting becomes available after making secondary missions for Cletus Trevor. When you begin hunting, you get an unlimited ammo to sniper rifle quad and a whistle to lure out nearby animals. Whistle (left direction button) you use to identify place nearby to which are animals. Then, you have to find a target and kill it. Cletus suggests sneaking in (L3) to minimize noise. But in practice, animals won't get scared unless you get real close. Use quad to move faster, especially to making photos. Look also at the wind indicator and move always against wind. Notice also indicator's color: when it's green you can walk steadily, when it's red turn away.
Your target is almost everything which moves, except the does (the one without horns). It can be a bird, coyote, deer or boar. Then, you have to photograph the animal and send your photo to Cletus. He evaluates it and transfer proper amount of money onto your account. The most precious are the deers, 5 points and 50$. You can increase it to 7 points and 75 dollars for hit in the neck and 10 points/100 dollars for hit in the heart. As the game progresses, you can receive special challenges (like kill two deers without a whistle or drive down the boar), which will give you more points. Hunting lasts to the twilight. Earned points determine the medal you get for hunting. When the day is getting to be over, you get a message from Cletus. Hunting may be resumed only at the next morning.
While hunting, you have to beware of cougars. Those closest ones are marked on the radar, you'll also hear a specific noise. Change the sniper rifle to shotgun. In case of need use your Special. When it's launched, Trevor will survive an attack and will kill the car when he gets up. Otherwise, he dies immediately. Killed cougar can also be photographed and send to Cletus.