This will happen quite often in this game but for newcomers who want a simple quick guide, here you go.
So you need to wound the animal, meaning shoot it somewhere where it wont die right away. Like the ass or stomach. After you shoot it, you will want to follow the blue trail. Approach where you shot your deer and you should see the deer's tracks along with some blood. Follow the blue track marks only as this is your deer. The white tracks indicate another deer and if you examine them, the blue tracks from the deer you shot will turn to white and you may end up losing your deer.
Follow the blue trail until you find your trophy. Now simply press X to claim the trophy and the cheevo is yours. That simple.