After you liberate an outpost, certain "jobs" will be posted on a bulletin board. These jobs will be given icons on your map if you apparently have trouble finding them.
To complete a Path of the Hunter quest, you will be sent to a specific area to hunt an animal. You will be given a weapon at each location which you are required to hunt that animal with. They are not timed quests. Some quests require you to kill more than 1 animal, so make sure you are prepared so walk the land quite a bit.
The trophy will unlock after you bag your last animal (or the only animal, depending on the quest).