Fighting the Four Kings, as I'm sure you know gathering from your prior experience, is a race against time. As far as melee goes, I really only think of two distinct approaches when it comes to the Four Kings.
Equip the absolute best armor you have, defense-wise. You don't have to care about how well you can roll, so feel free you equip yourself up to full burden. Next, equip the absolute best weapon you have for your character in terms of raw DPS (damage per second). Edit: I actually can't remember off the top of my head if the Four Kings have a damage type weakness, but I'm pretty sure they don't. I find that raw physical has worked well for me in the past.
When the fight starts, chomp a Green Blossom and sprint towards the first King as quickly as possible. Try to stop sprinting a few steps away so you have a good stamina bar once you're in melee range. Get as close as possible. Literally try to kiss the King. The closer you are, the less damage their sword swings do (or whatever those blades are supposed to be). At this point, start attacking the King, dealing damage as efficiently as possible (i.e. at a constant, steady pace, as fast as possible -- don't overdrain yourself on stamina so you have to stand around waiting a second or two, but go as fast as possible). If you've done this right, the King should die before the next one spawns. At this point, you just repeat the process.
You only have two attacks to watch out for: the one where he curls himself up and explodes with Dark, and the one where he grabs you. You'll just have to learn the telegraph for these and not get hit by them. The regular melee attacks will deal very little damage to you if you're positioning yourself correctly.
If you have trouble surviving, get a Pyromancy flame and Iron Skin. The problem is when you kill one of the Kings, you'll be stuck in Iron Skin, and get pelted by those devastating magic missiles by the next King. You could use Iron Skin on the last King, though.
Well, this is the same way every Dark Souls "pro" does it for every boss. This is basically the exact same procedure, but you equip minimum burden, and you have to roll dodge more. I don't know how good you are at using the game's invulnerability frames, but if you aren't, this will probably teach you. Hint: they're close to the beginning of the animation.
Maybe you could tell me what you've tried and what specific problems you're having in the comments for this answer?