I would like to add to Mumbo Jumbo UK's post by pointing out that if you shoot any barrels to kills off any soldiers, it does NOT count as a kill. Stupid as that may be, it does not add to your kill count. Also, as mentioned, doing this in Split-Screen allows you to get all the kills which on average can range from 50-60+. My personal record on this was 63 kills all in which only takes 2:10 per run.
I'm going to argue that this is the fastest way to get kills. The chapter checkpoint is called "The Fastest Way Down", in chapter 2.