The achievement seems to track kills across multiple playthroughs. You should get this simply by playing through a single campaign. Possibly 2 depending on the difficulty you have chosen.
It's also possible to grind the 500 kills in a single mission (although you will experience some significant lag as the body count increases).
One of the dark events you'll experience is a UFO that moves across the globe hunting the Avenger. By letting it attack the ship you'll be forced into a mission that requires you to destroy a relay while defending the Avenger's hanger doors. The benefit of doing the grind in this mission rather than a blacksite is some of your reserve troops will be deployed as well over time.
Focus on killing all the enemies on the map including those guarding the relay. The Advent will start sending in reinforcements every turn. Simply move your troops to the landing zone and go into overwatch. Rinse and repeat.