Provide an enemy player with a freeze cam of your smiling face.
By the smiling face, this achievement means the taunt for your Grenade Launcher. The demoman lifts up his protective skirt, yell "KABOOOOOOM!!!" and flashing the smiling face conveniently attached to his crotch. Much like any other taunt, you are vulnerable while doing it, and the achievement is acquired from the freezecam, not somebody actually taking the picture. It's borderline useless to farm this, as all you have to do is get a single kill with the Grenade Launcher and taunt.
Note: This achievement is both a reference to the alcoholic drink known as a Bloody Mary and the British exclamation of bloody (similar to the F-bomb in American vernacular, although perhaps a bit more acceptable.) The Demoman says bloody a lot, such as "Stand on the bloody point ye halfwits!" so it's appropriate.
Provide an enemy player with a freeze cam of you shaking your rump.
Another taunt achievement! This one is can only be acquired through use of the Scottish Resistance, as the default launcher will make the Demoman thumb his chest and throw out a sign. When you taunt with the Scottish Resistance, the Demoman leans over and shakes his rump at the camera, which makes a squeaking noise for some reason (possible due to cartoonish physics). Once against, farming or doing this achievement normally doesn't really matter as kills with the weapon required are common. You just have to actually have the Scottish Resistance in order to pull it off.
Note: The reference of the Second Eye is to a certain part of the Demoman's anatomy. As the Demoman is missing one of his eyes, his formerly third eye is now his second. Potentially gross, yes, but ultimately hilarious.
He Who Celt It
Use the Sticky Launcher to kill an enemy player via environmental damage.
This achievement can be acquired with the normal launcher or the Scottish Resistance, it doesn't really matter. Environmental damage is defined as falling, drowning, burning, or instant death, so this achievement is actually quite a bit easier than you think. Two sticky bombs are enough to almost kill someone as they run over it and will bounce them into the air. If they die when they hit the ground, you will get the achievement. You can also blow them off of cliffs, into spikes, or whatever strikes your fancy.
To farm this achievement, get a friend to stand still while you blow him up into the air. Simple as that. If you want to go for a double whammy, let the first launch have a Medic healing your target so that the Medic gets the achievement for preventing a fatal fall. The second time around, launch him into the air and watch him die. You can also try and get the Highland Fling as well, but it'll be very tricky to do both at once.
Note: The name is an obvious reference to the phrase "he who smelt it, dealt it," which is most often used in regards to somebody farting. The Celts were the native tribes of Britain that ended up being displaced to Ireland and Scotland by the invading Saxons.
Destroy 100 sticky bombs with the Scottish Resistance.
Another grind achievement! The Scottish Resistance is interesting in that instead of blowing bombs back, it just breaks them as if they were shot. This occurs to normal or Resistance sticky bombs, so it can be done either way. If you are making an assault in a normal game, simply watch out for stickies along walls or the ground. When you see them, destroy them with the Scottish Resistance. Destroy 100 of them, and you will get the achievement.
Interestingly enough, farming this achievement is incredibly painless. Have a friend play as a Demoman and load all of his stickies onto a single point. You can lay 8 normal stickies or 14 Scottish Resistance stickies, which makes Resistance stickies better. Destroy each batch. After 7-10 instance of this, you'll get the achievement. As a side note, this achievement has the exact same icon as the Heavy achievement Own The Means of Production.
Note: This achievement is a possible reference to the robbery of England's crown jewels by a gang during Edward I's invasion of Scotland. Irritated by Edward's continued extortion of the populace, the crown jewels of England were stolen. While they were recovered, it certainly jarred Edward I.
Sticky jump a really long way...
Doing this normally and farming it are completely the same, so I'll just cover them together. By now you should know how to sticky jump with easy. If you don't know how to crouch sticky jump, or uber jump, here is the reason to learn. Whenever you explode a sticky bomb beneath your feet while crouched, and then stand up, you will go significantly farther, both horizontally and vertically. It's similar to the advanced strafe rocket jumping of Quake 3 yore. Anyway, you must perform this with a two-bomb jump in order to get the achievement. God places to attempt it are jumping between the balconies in 2fort or in Badlands. You can even merge this achievement with The High Road, if you can pull it off correctly. There's also a bevy of special maps designed for advanced jumping techniques, with the most played being Skyscraper.
Note: The Scottish Highlands are an area that makes up northern and western Scotland. It was here that a traditional dance was developed on a targe called the Fling. That traditional Scottish dance you see whenever a movie or show wants to get across that something is Scottish? That's the Highland Fling.