XCOM 2

First, let's talk about the easy way that does involve a roll of the dice, but when it works, it works like a charm: repeater mods. If you're going on the mission where Bradford insists on going on the squad, you're guaranteed to meet an alien ruler. Give everyone on your squad repeater mods. Every time someone shoots his or her gun and gets a hit, you have a 5 to 15 percent chance (depending on the mod level) of executing the ruler dead and unlocking the achievement. If you don't get it on that mission, keep your repeaters equipped until you've met all three of the rulers.

Now, let's talk about other sound tactics that will improve your odds without relying solely on repeater percentages. There are a lot, and this is by no means an exhaustive list, but here are some:

- At the beginning of the game, check the box to play the Alien Hunters DLC. If you don't, the rulers will still attack you, but they will begin on their schedule, not yours. If you do, you can wait until you're ready to meet the Viper King.

- When you start meeting the rulers, take along a psi operative. His Stasis ability is HUGE because it allows you to position your squad, reload, heal, etc. without allowing the ruler a reaction every time. This is super huge when the Archon King charges his Blazing Funyuns attack. Slap him in stasis and let everyone underneath him get away.

- You need to take down the armor quickly. Acid grenades, plasma grenades, frag grenades, and the grenadier's Shredder ability all do this. A grenadier spec'd with Shred, Heavy Ordnance, Volatile Mix, and Salvo and supplied with acid and plasma grenades can get you off to a good start.

- Being able to deal persistent damage also helps a lot. Acid grenades, incendiary grenades, dragon rounds, poison rounds, etc. wear down the ruler every time he or she takes an action, not every time you take a turn, so their damage dealt is much higher than with ordinary enemies. Also, if the archon king grabs one of your soldiers while under a damage effect, he'll drop him automatically, so that's one less thing for you to worry about.

- Being able to deal guaranteed damage helps when the Viper King has bound one of your soldiers or the Archon King has taken one of them. Combat Protocol, Soulfire, and stock mods are ways of dealing guaranteed damage.

- Abilities that don't cost an action, such as the ranger's Ax Throw and the gunslinger's Lightning Hands, also don't cause a ruler reaction.

- If the ruler is the only enemy on the screen and you don't have any good moves left - only low-percentage shots from weak weapons - end your turn using the Menu button on your controller (what used to be the Back button on the 360 controller). The ruler will get an "Alien Activity" turn against you, but the results are no different from what they would be from a "Ruler Reaction" turn, and you'll be able to use your heavy hitters again.

In the end, though, if you use repeater mods, you're probably going to unlock the achievement with them instead of by any other tactics one can come up with.

EDIT: On a new Legend playthrough, I killed the Viper King the first time I met him in Vahlen's cave. As soon as he came out, I lobbed an acid grenade on him, and applied Rupture to him on my second attack. After my third attack, he had already had enough and tried to bail. He opened a portal, but then he didn't use it. Instead, after my next attack, he closed that portal and opened another one. He did the same thing on my next attack. I'm not sure, but I think the pool of acid surrounding him (and causing 4 damage with every reaction) was preventing him from using the portals he was opening. I was able to kill him not just in our first meeting, but in my first turn of our first meeting. Acid grenades!

WOTC UPDATE: One tactic available in War of the Chosen is to use the Reaper hero's Banish ability, unlocked at Major. You can combine this with Blood Trail (corporal) and Needle (lieutenant) for even greater effectiveness. Be sure to position your Reaper on high ground and equip his or her weapon with an expanded magazine or auto-loader. If you want more control over when you're going to meet the rulers, uncheck the "Integraded Downloadable Content" box in game setup and check "Alien Hunters: The Nest." Doing that will give you the familiar "Alien Nest" mission that you can start whenever you're ready to take on the Viper King.

I would suggest waiting until you have plasma weapons before starting the first Alien Hunters mission.
What I did was equip my squad of 6 with plasma (except Central of course) and a cryo grenade. I went with
1 Sharpshooter
1 Grenadier
2 Specialists
1 Ranger

The grenadier had shredder to lower the target's armor and allow for maximum damage. Once the target opened the portal to escape I hit him with the cryo grenade and continued to unload on him. For the most part, I ignored the adds.

Updated 04/10/17
I got this on the first encounter. Its important to bring the bolt caster for my strategy.

The encounter begins when you get close enough to the transponder, its past the cliff and on the ice.

I was at mag weapons level on the lowest difficulty. My trick was to get all 6 members on the high ground above and then trigger the Viper King. I then shredded as much as possible and once he opened the portal used lightning hands as he doesn't react. The trick was to use the bolt caster to get a stun for two turns and then give it everything you have!

You can overwatch trap before you trigger the Ruler and get 5 actions off at once whereas normally they get an action for every one of yours.

Update for WotC - Banish is an extremely useful perk on Reapers which you should make use of if you can, maximise its effectiveness by using an superior extended magazine and superior repeater. If you can equip repeaters on as many people as possible an execution can also be what you need.

WotC - Berserker Queen - Use of a Frost Bomb, Rupture and Banish.
http://xboxdvr.com/gamer/Vr%20English/video/37394255

First Encounter - Viper King - Overwatch Trap and Repeater Kill
http://xboxdvr.com/gamer/Vr%20English/video/37394362

War of the Chosen has opened up another reasonably easy option for this achievement. The Reaper hero unit (you'll get "Outrider" for free after completing the "Lost and Abandoned" mission) has access to the Banish ability once they reach the rank of major. Banish allows the Reaper to continue to fire at a target until it's dead or they run out of ammo. (There's also an upgraded version at colonel rank but that's not a requirement).

So it's just a matter of making sure you have your Reaper with you when you're expecting to run into a ruler - generally when assaulting an avatar facility - and having a full clip good to go. You can tip this in your favour by upgrading their vektor with either an auto-loader or an extended magazine (either will work but the extended mag is probably better).

On the off chance this doesn't kill the ruler outright, it should be weak enough that conventional attacks by the rest of your squad will finish it off. Alternately, you can soften up the ruler before you unleash the Banish attack just to make sure (any armour shredding is a plus).