Holster your gun immediately if you are going for this achievement. You do not have to take down the Influenced players, just scan the evidence. Work on your blocking and counterstriking. You must do this for 10 RANKED rounds. You WILL get credit for the round if your partner shoots but you do not, and you still win the round.
I've heard people say "Throw your gun away". That just means there is now a gun for the Influenced to pick up. Grab a good melee weapon and just don't fire a shot.