For this one, you have to be the one player out of both teams to achieve the most kills in a game. Don't let the picture fool you, the icon for Vanquish Master is the purple skull. For some reason, there are a couple of the boast cheevos that were mislabeled with the incorrect icon. The icon for this achievement actually goes along with the Assist Master Boast.
It's hard to say which class is best for this, as I have had rounds where I've gotten this boast from playing each class on each team. The All-Star is a good choice on the zombie team as he has the mini gun, sprint tackle, and the imp bomb. All those make a good set up for a bunch of multi kills and a ton of damage dealt to the plant team. As for the Plants, I would overall recommend the Pea shooter mainly for the Turret and Chili bomb. Being in well placed areas, the turret can rack in some major kills and the chili bomb has a decent range and can provide some quick single or multi kills.
Hope this guide helps and happy hunting!