First unlock the Passive ability for gaining Adrenaline every time you make a successful dodge. Remember to only dodge when they charge. You should see a red marker when they are about to charge. Just keep dodging until it says Press X for Adrenaline. Don't shoot at the animal until you get this.
Now for the achievement. Buffalo
Region 1 - This is found early on in the game and a good place to get this, is at the end of the region. (Boss fight)Elephant
Region 3 - There will be a mission near the start of region 3 where you will have to fight a lot of elephants. The best way to kill an elephant is to dodge it and shoot it's side.Hippopotamus
Region 3 - These might be kind of hard to find and dodge. The best way to find a Hippo is to walk on foot and not drive. Go through the swamps and you will find one eventually. Now this may take a few tries but I found the Hippo to be the 2nd hardest to dodge. Just keep rolling when it runs at you. I think it charges when it turns around. So run behind it quickly and wait for it to turn and then hope it charges and dodge.Rhino
Region 4 - There will be a few missions where you get to fight a rhino. As long as you dodge one of them, you will get the achievement.