This is achieved on Mission 19 (Dam). This mission lasts roughly 5 minutes and is very easy to complete. You cannot equip any weapon throughout the level, but you're allowed to use mounted guns on vehicles and any found in the environment. If you do pick up any weapons by mistake throughout the level, simply restart the checkpoint as this will not void the achievement.

As soon as the level starts, be careful not to accidentally pick up the M4 rifle on the ground just a few feet ahead to your right. Run forward, past the trigon enemy and jump off the waterfall, follow the river until the cutscene plays. Now make sure to grab the armour on the ground and use the mounted gun inside the fallen chopper to kill all trigon's ahead of you, one enemy will also spawn behind you after everyone is dead ahead, simply switch sides to the other mounted gun and kill it.

At this point, you will want to make your way up the hill towards the big gate, sticking to the right to avoid most of the enemies, go through the fort and to the right into the armoury to grab the keycard and armour if you need it. Make sure not to accidentally pick up any weapons in the armoury. Now go back outside, open the gate and into the jeep. Drive down until you reach the lava streams. You will have to cross the first stream by walking over a fallen tree acting as a bridge. You the cross the second stream by walking across some rocks further up the hill.

Now run past the enemies, carefully slide down the cliff to make sure you don't lose too much health when you land on the bottom. Make your way towards the large structure on the right, up the stairs and to large door guarded by several mercenaries. You will most likely die at this point but the level will also end and the achievement will unlock.

See video for further help: