I did this on Legendary. As I'm rather rubbish at Legendary, I added Famine (as there's no shortage of dropped weapons, so ammo will not be an issue) and Iron (as I was using the Home Screen-Quit method to avoid losing points on dying) to give a multiplier of 6.24.
I took my time going through, trying to strike a balance between killing everything and not getting my butt shot off, reloading before the death animation completed if I got killed. Also, once I got the vehicle, I stuck with it until the Falcons came to pick everyone up - that thing can take a lot more hits than I can on Legendary, and it was fun getting splatter multikills on the Grunts.
In the next srction, I stayed back and sniped with the DMR through the door until it shut, and then tried to kill all the Grunts and Jackals in the night vision section before Jorge did.
I didn't use a plasma pistol on the final Sword Elite, instead I dropped down off the ledge for the shotgun.
In the end, I came in well outside of par time, but with just enough points for the achievement to pop.