Defeat the White Defender. This is the dream version of the Dung Defender. The 3 (possibly 4) requirements are to have the Dream Nail, have defeated the Dung Defender, and have Desolate Dive or Descending Dark (the 2 ground pound spells). As of Aug 9, 2017, the Hollow Knight wikia says you must also defeat the 3 dreamers (Monomom, Lurien, and Herrah) before you can fight the White Defender, but user Bloodycrow from the comments section states it can be done without defeating all 3.

When ready to challenge the White Defender, go to the area immediately right of where you fought the Dung Defender. You should see the switch above you that drained the sewers. Stand on the ground directly below the switch (and under the small, floating platform as well) and use Desolate Dive/Descending Dark. You should break through the floor even if there isn't the normal indication of the ground shaking underneath like most other breakable floors. Dung Defender's body should be somewhere in this new area. Use the Dream Nail on him and be prepared for a harder version of the original Dung Defender fight!

Note, it is in the dream dimension (like some of the other boss fights), so the 3 Fragile charms won't break if you die, you won't lose geo, you won't have to reclaim your Shade, and you can't fail in the Steel Soul mode!

You only need to defeat him once for the achievement, but if you're interested, you can fight more iterations of him.

Defeat the Grey Prince Zote. In order to access this fight, you'll need to have done several things in your playthrough. Save Zote everytime you've come across him in your journey (which is more than just the first time you come across him in Greenpath). Then defeat him in The Colosseum of Fools (in the Trial of the Warrior which is the first unlocked challenge in the colosseum). You also need to have saved/talked to Bretta in the Fungal Wastes, so that she will return to Dirtmouth. With all this done, Zote and Bretta should be in Dirtmouth outside her house. Go inside her house and descend to her basement. Use your Dream Nail to enter Bretta's fantasy of Zote. You'll fight a much improved version of Zote.

Note that the fight takes place in the dream dimension, so death will not cause the Fragile charms to break, geo loss, Shade reclaiming, or the end of the Steel Soul mode.

You only need to defeat him once for the achievement, but if you're interested, you can keep fighting stronger iterations of this fantasy Zote.