This took me about 20 minutes of play to achieve (although I've played the game plenty before, so know how it all works), so it should be fairly straight forward to achieve even if you're new to the game.
If you look at the wiki (or the XML file, or the in-game crafting menu) you will see that Diggle Nog can be crafted using a Porta-Still and combination of Aqua Vitae, a Diggle Egg and some Grog.
Therefore, to get the achievement you need to find all three craft items and a Porta-Still to combine them with. Craft them together and that's your achievement.
So, to minimize the time to get this achievement I'd highly recommend starting with the Alchemy skill so that you start with some Aqua Vitae (which will probably be in your quick bar, not your inventory) and a Porta-Still. You'll then need to find the other two items yourself, but they're fairly common items that you should be able to find somewhere on the first level.
Keep on the lookout for Drink Machines:
If you take Burglary as a skill, you can steal random liquor from drink machines, which may speed up your acquisition of the Grog. Otherwise, check each one you find to see if they're selling any.
Diggle Eggs are found frequently in Diggle Nests, which are small rooms containing a couple of Diggles and usually nests filled with eggs:
Waiting for an ideal random placement is part of it (I was fortunate enough to have 4 prisoners located nearly adjacent to each other. A pity it took me so long to find them!), but the other thing is that you don't have to kill everything on your way.
Speed is of the essence, so skip combat when you can. Abuse the crap outta Dashing Strike and/or Tempest Rush (You're playing a Monk, right?) (Or whatever movement skill your class has) to get in, get out, and keep going.
And lastly, don't give up if it takes a few goes. One run the random numbers will fall into your favor, and then you'll get it in the end.
Best Answer
Based on a rough timeline of the data I collected, there are 23 points during the game that have to be hit to complete all the conversation achievements and, because of dependencies, they need to be hit in order. When talking to someone, you are expected to complete all their available conversations before moving on to talk to someone else, but I only count the conversations that count towards achievements despite there being additional extra conversations available. I will try to group conversations to cut down travel time, making this pretty close to the fewest stops with the least travel, but there is probably some room for optimization once all dependencies are mapped out. Don't worry about avoiding people with asterisks because you will get those conversations later and this way involves less running around. I'll go through and confirm specifics and nuances later to determine if there are inconsistencies to timing and number of conversations based on current objective and dependencies.
Act I
Act II
Act III
Act IV