The requirements of this are pretty straight-forward, but one may hasten getting to 40 by saving a game right before moving to the final battle (save while at your Mission Control), load up that save, and go to the Mission Control screen. Start to "Scan for Activity" -- if the result is anything but a "UFO Contact," skip through it without accepting any missions, and go right back to scanning for more activity.
Doing this, you'll eventually start to lose council nations to panic, but you're just trying to rack up some quick shoot-downs with your Fusion Lance-equipped Firestorms.
You should be able to get around 13-15 additional kills towards your total goal of 40 (across all games) before the 8th nation leaves the council, assuming you started with all nations still in the council (so, you'll also get to see the 'losing' cutscene, which you probably hadn't seen, yet).
Important: 'Enemy Unknown' and 'Enemy Within' do NOT add up. If you got 20 on Enemy Unknown and another 20 in Enemy Within the achievement won't pop.