Dominate three Engineers.
This achievement is much easier than it seems. While Engineers are certain a formidable force, and any kill from their sentry translates to a kill for them, they are also prime targets for a good Demoman. They set up their little nests, and these are prime for your sticky bombs to blow to smithereens. Just do this over and over while avoiding death by the hands of the Engineer, and you will net yourself a domination. Dominate three times, and you have yourself this achievement. It's as easy as that!
To do this quickly, get three Engineer friends to stand on a pile of stickies as you detonate them. Do this three times to get this achievement. If you want to, you can have the Engineers also build up their nests, which can get you several other Engineer-related achievements, such as The Argyle Sap, Blind Fire, Loch Ness Bombster, or Cry Some Moor!
Note: The three dominations do not have to be in a single life. They don't even have to be different people. If you dominate an Engineer and he gets revenge on you, he's fair game again. This means you can do this achievement with only one other person if the need be.
Kill an Engineer building that you can't see with a direct hit from your Grenade Launcher.
This achievement is alternatively very easy and very hard. The gist of it is that you have to bounce a grenade around a corner to kill a sentry with a direct hit. While this is easy by itself, there will also be an Engineer repairing and defending the sentry from attacks, which is what makes this achievement difficult. Your best bet is to attack the sentry when it is a low level while the Engineer is distracted. This is especially easy if the Engineer keeps running off to get more metal from drops or ammo boxes. Once you kill it with a direct hit, you'll get the achievement.
To farm, have a friend build a sentry gun near a corner where you can rebound your shots to destroy it. Then do so. You now have the achievement! Easy achievement to farm and great to do alongside other achievements such as Three Times a Laddy or Laddy Mcdeth. You can weaken the sentry with a sticky grenade or two before you destroy it, if you want.
Note: You have to directly hit the sentry gun, as mentioned. However, you don't have to bounce the grenade. You can also shoot it from on top of a platform down onto the sentry. It's normally easier to do it via bouncing, though.
The Scottish Play
Get a melee kill while sticky jumping.
Quite difficult in normal play, this achievement functions under the same restrictions that the Soldier's Duty Bound achievement does. You can land, but it has to be on an enemy who you then kill almost immediately. You have to be launched via your own sticky jump, not launched through damage taken from a rocket or another Demoman's stickies. A nice change is that you can use the Eyelander or bottle on this achievement, as opposed to just the Equalizer for the Soldier's version of the achievement.
To farm this achievement, get a Soldier to rocket jump at you while you sticky jump at him. Then give him a good whack with your Eyelander. Make sure he is using the Equalizer, as this will give you the Slammy Slayvis Woundya achievement as well. It's a fairly easy achievement to grind, especially if you have the Soldier damage himself once before rocket jumping up to you.
Note: The Scottish Play is a euphemism for Macbeth, which is referenced constantly in the WAR Update. Of note is that the play has a superstition around it where saying Macbeth causes massive bad luck for the actor involved, and the actor must cleanse themselves with a traditional theatrical ritual. Kind of strange if you really think about it!
The Argyle Sap
Blow up an Engineer, his sentry gun, and his dispenser with a single Sticky bomb detonation.
A very easy achievement, and one of the first you should get, The Argyle Sap involves destroying an Engineer nest. Those nests are a nasty infestation, especially if they are in your own base, and you are the cure for such an occasion. Simply launch some stickies (5 or so should do it) into a nest. Then detonate them. Once you do so, you'll destroy the buildings, kill the Engineer, and get this achievement. Note that you do not have to destroy a teleport for this achievement. Just a sentry, dispenser, and the Engineer himself.
Farming this achievement is just like the rest of the Engineer achievements. Have an Engineer build his sentry, dispenser, and teleports in a position where you can plant stickies on them. Alternatively, plant stickies and have the Engineer build on top of the stickies. Once he is done, detonate them. If you have him travel through the teleport, you'll get Loch Ness Bombster, and the four building destructions will count towards Cry Some Moor! Farming heaven for this one!
Note: Argyle is a type of pattern with interlocking diamonds set in a checkerboard pattern. The pattern is derived from the tartan of the Campbell Blan of the Scottish Highlands. The Demoman is Scottish. And thus, the link between the two is formed.
Slammy Slayvis Woundya
Decapitate an enemy Soldier who is brandishing the Equalizer.
This achievement doesn't really matter if you farm it or do it normally. Here's all you do: kill a Soldier who has the Equalizer with your Eyelander. That's it. There's nothing else to it. As the SOldier will likely be low on health when he has the Equalizer, this is a fairly easy achievement to do. You can also use the Chargin' Targe to snag this achievement, as a critical will guarantee a decapitation on an Equalizer-wielding Soldier regardless of his health.
Note: This achievement is a reference (and pun) to Sammy Davis, Jr, one of the members of Frank Sinatra's Rat Pack and a prolific performing artist in theatre, movies, and music.