Credit for this solution goes to spockroxfaces from trueachievements. They said:"
"The key to getting this achievement is first knowing that the xbox tracker is broken. During the 2nd DLC mission of Enter the Dominatrix you'll encounter the Dominatrix right outside of a friendly fire. Attack her until she disappears and then go into the friendly fire and equip whichever of the 3 weapons, flamethrower, minigun and grenade launcher that you want to work on. If you stay in this area, instead of continuing the mission, you'll get an endless supply of incoming enemies. Blowing up the vehicles that your enemies are in counts towards the total and getting all 3 achievements should take about 10-15 minutes. Make sure that you fully upgrade each weapon and go to town. TRUST ME WHEN I SAY THAT EVEN THOUGH THEY AREN'T BEING SHOWN ON THE TRACKER, YOU WILL STILL GET CREDIT."
I can personally confirm the kills count when you destroy the vehicle and get an "explosive kill". There is no kill counter, so you just have to kill until you get the achievement. I'm not sure if kills count outside of the DLC missions.