For this achievement you need to complete the game using only the trophy gun and Compound bow. After you complete the game for the first time you will unlock the Trophy gun. You cannot use any other weapon which includes the shotgun, rifle, and long range rifle. You cannot for instance use the shotgun to hunt birds as this will disable the achievement.
So you need to go on to level select and play through the missions using only the trophy gun and compound bow. Try to remember which levels you have done because it starts you at the last level every time you complete a level. You can replay the levels in any order and if you do use another gun for a level don't worry just replay that level and the achievement will unlock. I for instance hunted birds with a shotgun for the first two levels and had to replay them for the achievement.
To get the kills for the small and animals and birds just use the methods I have outlined in the guide and not during this playthrough it will be easier that way.