This can only be done on a crash site mission and requires a single soldier to kill all aliens on the map. Although it can be completed on ether Classic or Impossible difficulty, it will be much easier on Classic.
The easiest mission to achieve this will be the Small Scout crash sites, as they only have anywhere between 5-8 enemies. Small Scout UFOs will appear early in the game, then randomly towards the end. If you choose to attempt this late in the game, then any or your upgraded soldiers with decent weapons and armor shouldn’t prove too difficult to clear the map on their own.
An alternate strategy is to do this on the very first crash site mission as this is guaranteed to have 4 Sectoids and an outsider. Start a new game, disable the tutorial as this gives you better odds of one soldier being chosen as an Assault. Play through the 2nd mission (abduction) focusing on ranking up your Assault soldier to corporal, (or Squaddie if you missed out on the first mission) and at the end choose the ‘Tactical Sense’ perk. An alternate to the assault class is the sniper utilising the Snap Shot and Headshot ability.
Make sure to save before you deploy to the crash site. (This will mean that every time you reload you receive a different map but this shouldn’t make a big difference to the strategy.) Once on the ground, move up one side of the map slowly. Ideally you want to engage the Sectoids two at a time. With a little luck, 1 of the 2 Sectoids you uncover will mind merge with the other. If this is the case use Run and Gun to move into a position to take out the one doing the mind merging, so that by killing it, you also take out the 2nd Sectoid. If using a sniper, be sure to use headshot on the mind merging Sectoid or take out the Sectoid posing the greatest threat first.
Be sure to reload, and hunker down for 2 turns to recharge your Run and Gun, then repeat for the other 2 Sectoids. Sometimes you won’t be able to get close enough to make use of Run and Gun, so fall back on moving once and using a grenade. Also sometime the Sectoids won’t mind merge, so it is best to fall back and engage them one by one.
Once you have cleared the Sectoids, move up to the UFO and take up position at the front near the door. Do not approach from the side as this can often mean the Outsider will spawn on your second move. Once you are near the door, end your turn, then open the door to make the Outsider spawn. Chances are it will run directly for you. Move inside the UFO giving yourself a clear shot and take out the Outsider. The achievement will unlock on the mission results screen.
This achievement relies heavily on luck so it may take several times to get the right alien placement/strategies before it works in your favor.
NOTE: Some people have reported this achievement to be glitched and will not unlock if you reload a save at any point during the mission. A good strategy is to make a save before leaving the base and reload from that point if you die during the mission.