All you need to do is hunt 300 birds with a shotgun in story mode. This will more than likely take two playthroughs. I would suggest you do the method below because on your second playthrough you will be going for the Gun Master achievement and using the shotgun disables the achievement.
Here is how to boost it really fast: load up Texas day 1 hunt 2. You will come to a open area with fences on both sides. You will be able to shoot about 35-50 birds each time. After you kill the birds just walk forward a little and shoot the trophy boar and you will get an auto save. Then just quit to the main menu and choose this level again. Only takes about 15-20 minutes to get 200 birds killed. Also make sure you use your shotgun as kills with the other guns don't count.