You have to not open any chests in an entire level. This means not just walking on top of them but also not shooting them. Chests include the obvious treasure chests with weapons in them but also the orange ammo box and blue experience box too. Just keep trying on level 1-1 and you will eventually get a small map with only a couple of chests. This is shown at 3:49 in the video.
Full video achievement walkthrough: