Requirements:
•Reinforced Estus Flask which is kindled. (Recommend the Rite of Kindling as you can take quite a bit of damage from some of the attacks)
•Crest Shield (High magic resistance is needed for the kings)
•A powerful weapon able to kill a kind without too many respawning. (Black Knight sword +5 worked great for me)
•Ring of Favour and Protection.
•Be moving at normal speed howver nimble is not required.

Walkthrough:
As soon as you fall down into the abyss run directly to where the king will spawn. It helps if you have already died as you bloodstain will show the direction where he spawns. Running directly to him will lower the chance of him doing magic attacks.

Suprisingly enough I found the best tactic for taking out the four kings is to just stay as close as you can to them. Since their attacks do more damage the further away you are. This will make the damage they do to you negligable.

Just attack and let their attacks hit you. Try to strafe around them whilst attacking as this will make quite a few of their attacks miss. The only attack you need to worry about is the thrust attack. This attack can do a load of damage to you even if you are standing right next to them. When you see him preparing to thrust either dodge or block.

Also block against the horizontal swipes as attack still does quite a bit of damage even at close range.

The dual diagonal swipe is not a threat at all as when he does the first one just move to the left and carry on attacking and the second swipe should miss.

If you stay at very close range the kings should do their grab attack much less frequently however if they do just carry on attacking as you will more than likely be too close in to dodge it.

When the king is doing it's dying animation carry on attacking it as health will still be taken off the total health.

As soon as the kings dying animation is over run straight to the next king and repeat.

Tips:
•The best times to heal are when he is recovering from either a grab attack or a magical blast attack. You can also heal after you kill one of the kings but keep an eye on the other ones so you don't get swarmed.
•If a king spawns whilst you are still attacking the other do not worry. If you stay close enough to one king the others should keep their distance however they may do a swing at you or do a ranged attack but this is uncommon.
•Do not waste Estus Flasks and Trainsient curses getting there just do the shortcut ladder by the stairs and jump off the building to the boss fog gate.

(sorry for no video as I did not really expect to kill the kings with this stragety but it worked very well)