This is the hardest achievement in the game. Quick Draw galleries require you to get to a certain level of points, and provide you with perks and point multipliers to make it easier. The idea is to reach the maximum perk level in a certain amount of time. Unfortunately, if you get hit, you will lose the most recent perk you acquired and have to start again at the beginning of the previous perk, though you will keep your multiplier. You have very little margin for error, so you are basically going to want to not get hit at all, meaning that predators or any animal charging at the screen will always be your top priority. You will also probably need a few runthroughs to give you an idea of where the predators come from. Tips can be found HERE.
The linked thread suggests playing on hard for the Quick Draw galleries, but medium may be easier. You will get fewer points, but it will also take fewer shots to kill the predators, giving you more margin for error.