This achievement is for beating the Lernean Hydra. He can be found in the Athens Marsh in Act 1, which is a subarea of the "Monster Encampment". You go through the Monster Encampment on the way to Athens (it's between the Parnassus teleporter and the Athens one). Once you reach the Monster encampment area, simply refer to this map. He only appears on legendary difficulty.
The main resistances you want are fire and poison, and cold would be nice, but not needed as much. He has your regular melee attacks and 3 breath attacks, depending on which head breathes. It can be fire, ice, or poison. The fire breath does pretty good damage. The poison breath one does the most. The cold one seemed to do the least damage to me, but it also applies a slow, which can make it easier for the Hydra to hit you with other attacks, so it can't be fully ignored. His last attack uses all 8 heads to bite you in melee range. The wiki says it also does poison damage, and it's the strongest attack he has, even worse than the poison breath. If you're melee, when I tried to run around him the multi-bite attack would always hit me, so I assume if you want to melee him you'll need to max out fire and poison resistance (and some cold) to be safe and bring some life leech, so you can just stand and deliver through his damage instead of trying to be fancy and dodging stuff.
I used ranged (bow Haruspex). If you're ranged the fight is easier and you won't need max resists, though make sure you clear out the monsters near his lair so you have room to kite. Run away from him (as in opposite the direction he is) until he uses a breath attack. Keep running for 1/2 of a second to a second, then hit him with some arrows (or spells). Repeat until he dies. The hardest part is making sure you run far enough away after he uses his breath attack to make sure it doesn't hit you before you start shooting him, but stay close enough you don't lose the autolock. Once you get the spacing down it's entirely possible to do the fight without taking damage once, unless you let the Hydra get into melee range. I found that when I tried to run around him the breath would sometimes hit me, which is why I recommend always running away from him instead of around him. Kite him in a very big circle, basically.
The achievement will pop once the Hydra dies.