This battle takes place in the Abyss, which is accessible from a tower at the bottom of the New Londo Ruins.
The Four Kings are one of four primary bosses that you must defeat to open the path to the Kiln of the First Flame which leads to Gwyn, the game's final boss. Take the lift below the Firelink Shrine bonfire to get to New Londo Ruins and cross a rickety wooden bridge to get into the actual area.
This area is full of ghost enemies that are initially immune to all damage. You can circumvent this hindrance by using a Transient Curse, which lasts five minutes and allows you to deal damage to ghosts. Make your way through this linear area to reach a large building with a ladder to the roof where you will meet Ingward, who will give you the Key to the Seal. Progress farther to find a door that this key can unlock and a lever behind it that can be pushed to drain water out of the area.
You can now access the lower part of the New Londo Ruins where you will encounter Darkwraith enemies that are pretty tough.
Progress farther through this area to reach a fog wall that leads into a circular tower with steps going down. You must equip the Covenant of Artorias in one of your ring slots (gotten by killing Sif, the Great Grey Wolf) to proceed down into the Abyss or you will instantly die. Drop off of the steps into the darkness below and the battle will begin.
The first of Four (or even more) Kings will appear shortly after you land.
The Kings have several different attacks. Time is of the essence in this battle because subsequent Kings spawn on a timer and engaging more than one at time is nearly impossible. Do your best to damage each King, who's individual life bar is about 1/4 of the main boss life bar. After you kill a King, keep attacking it as it fades away as this will allow you to do extra damage to the main boss life bar.
Ideally you can kill each before the next spawns. You will need to kill at least four in order to deal enough damage to the main boss life bar, and Kings will keep spawning in beyond a fourth or even a fifth if you aren't fast enough. Once you completely deplete the main boss life bar, the battle will end (with a brief cutscene) and the achievement will unlock. Good luck!
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