This is an easy achievement but misleading. What you need to do is complete the entire level without firing your gun, not just the cave portion. I will note that after the Cave part, you will be put into a hospital looking area and this will contain a statue with a key that requires a gun to get. You will not be able to use your gun to get this. Alternatively you can use a bottle that is on the floor in that room.
As for how to accomplish this. Plain and simple, run through the enemies. In the first bit what you need to do is run past the group in the first area, you can make it easier on yourself by luring them back to the starting position. In the next room will be a gate with two enemies. You will need to lure these two out of that room and take them to the starting position just like the group above. The reason I recommend luring the first group is that the passage is narrow to the 2nd room so if one or more follow you, you are stuck.
Once you lure them out and raise the gate. You will be able to run through the chapter until you reach the next room with a gate. Once you start opening the gate, little monsters will rain down on you. If you are lucky, you can lift the gate without any of those monsters attacking you. If not, just running over them will kill them. After that it is pretty much smooth sailing. You will be able to run by any enemies after that point.