I was struggling a bit getting this achievement the normal way with guns and being cautious. The way I got it was using the Slasher class. You need it to be at least level 17. This way you can buy the perk:
Heavy Metal I. The character starts with an improved Forrest Warrior Chainsaw, dealing 50% more damage and having 50% more fuel. All Forest Warrior Chainsaw pickups are also improved.
Now equip it from the start and only use your chainsaw the whole level. You can chainsaw a lot of zombies and slash your way through groups without hitting your teammates, because this is disabled by default! The improved version of the chainsaw gives you a lot more fuel to maintain the weapon for a much longer time. At least untill you get a new one somewhere in the level.
Extra Tip: Use the LT over the RT attack, because this kills a zombie with the least ammout of fuel.