This achievement is probably most easily done on the 2nd ruler you encounter.
*** Spoiler - click to reveal ***The Berkserker Queen is a melee boss who runs at you so it'll most likely be in range of most of your soldiers. Any soldiers that can strip armor is very helpful to make each shot deal more damage, as is a soldier with the Rupture ability. Repeaters might be disadvantageous unless you make sure to not overwatch/shoot with them until after it starts its retreat. Try to have a grenadier with the frost bomb in the grenade slot in order to get 2 uses of frozen which gives 4 soldier moves of frozen for you. Damage and Strip the berserkers armor as best you can freezing it with a frost bomb or serpent armor ability if it gets close to your soldiers. Whittle the ruler down until he opens a rift to retreat. Try not to waste turns on relocating/reloading/overwatching while doing this, there is a manual end turn button in the top right you can use to end turns if a soldier's actions aren't worth giving the ruler another action. When he opens the rift put a nearby soldier on overwatch, or put a sniper killzone in his path and if he gets killed by the shot the achievement should pop. I'm not sure if a DoT like fire or poison killing it would count as a killing shot would.