This is actually a simple achievement to unlock. What you need to do, is during our turn purchase the Attack power up 4 times using Acorns. They don't increase in price either, making this easier.
I got this on the second mission. When you see a square with a pile of Acorns on it, place one of your cards down, be it a Squirrel or Barrel and you will collect them. You only need 800 acorns to achieve this, and it will pop once you purchase the 4th upgrade.
I'm not sure if the Acorn tiles are random or set. But just keep collecting them and you will soon have enough to go ahead with this achievement.
Another way to gain some Acorns is to combine 3 of a kind with your cards. To do this, just place them on adjacent tiles.