First, to find the City of Suffering DLC you need to travel to the Triglav subsector, Malcorum system, and the planet Myrkon. Note that you will need a power level of 1300 to unlock the Triglav subsector.
The entire DLC consists of a single mission on a single map that you can play repeatedly where you'll face off against an endless supply of Drukhari enemies until the mission ends.
This is an excellent mission for farming legendary items, by the way. You will receive rare loot boxes for completing the mission, and the higher your score the more of them you get, and virtually every one will have a legendary item in it. I was getting 4 loot boxes per completion when I finished in the range of 1200-1500 score. Nice!
Here's how it works:
When the mission begins, you'll notice a red bar on the right side of your screen. The red bar fills continuously as long as there is at least one pain catalyst on the map. When the bar fills completely the mission ends. Pain catalysts are the alien looking tower-like structures that buff the Drukhari enemies. Pain catalysts are indicated by a purple circle on your mini-map. Whenever you see a purple objective indicator on the edge of your mini-map, drop everything and hurry there and destroy it.
Destroy 100 of them to unlock this achievement. Your total is cumulative, so just play as many times as it takes to destroy 100 of them. The achievement will unlock immediately upon destroying the 100th pain catalyst.
If, like me, you're having trouble getting a decent score then learn from my pain and check out the solution for the other achievement in this DLC:
The Emperor protects!