Walkthrough 1 (Standard Entrance)
Entering Lost Izalith
Upon entering from the Centipede Demon boss arena, you'll find a bonfire on your left. If you've correctly followed Knight Solaire's quest line thus far, you'll also find him sitting beside this bonfire mumbling to himself. Continue onwards and you'll soon come across a big lava filled area dotted with ruined buildings, a few roots and other small pieces of land. You may also notice the many Bounding Demons littering the area, but I'll get to them in a moment. Before you set out into this area, put on the Orange Charred Ring that you got from the Centipede Demon to make you more impervious to lava damage - there is a relatively safe path through most of the field via the roots that cross it, but you will need this ring equipped at certain points to survive a brief walk through the lava.
Note: While the Orange Charred Ring will reduce the damage you receive from lava, your equipped armour will still degrade at the usual rate from lava damage, so ensure you repair your armour when you can after walking through lava.
The **Bounding Demons** are huge, fast and hit hard, but luckily they can damage each other and don't respawn. Even luckier is that they are also fairly non-aggressive unless you either get close or attack them, so there are various ways to lure one out at a time for the kill or, if you prefer, you can also entirely ignore most of them. To take them on, a good bow and a full stock of arrows is recommend as long as you can find a safe place to stand where they can't reach you. There are various places in the field where they will have difficulty reaching you; such as the entrance to the field just past the first bonfire (stand up on the root on the left side and lure them in with arrows) and also inside and on top of the towers. Take out as many as you can at range before you enter the field, and then start exploring. If you have no arrows or want to avoid fighting them at all then it's recommended to follow the path indicated by the roots.
Crossing the Lava Field
N.B. Although the lava itself will only do about 3 damage per tick, straying off the route is extremely dangerous due to the bounding demons. Also, one's armour degrades at a rather alarming rate, potentially cost a fortune in repair.
Following the roots across the lava toward the first tower you should find a soul item around a column that's on route (no pun intended). Once you reach the first tower you'll run into some Stone Demons and maybe a Bounding Demon as you circle around it. On one of the tower's sides you'll find an opening leading inside to a chest. Inside the chest is a Soul of a Great Hero, and you can also use this sheltered area to shoot arrows at any other nearby Bounding Demons safely. From here, rejoin and continue along the root path; which will lead up to a ledge and a broken bridge on the second tower. ( Tip: This is a great sniper spot to take out more of the Bounding Demons [what I've found to be most effective is to locate the two items while on top of the tower; you'll see them off in the distance while standing on the opposite side of the tower where the broken bridge is, along with the corpse holding Twin Humanities. now, equip a bow, and start shooting as many Bounding Demons as you can (don't worry, they can't touch you while you're up here; they'll swing their tails and jump real high, but you have nothing to worry about). what will happen is, they will group together trying to get you, while at the same time dealing damage to one another with their attacks; i took out 10 of them in less than a few minutes. once the coast is clear, drop down, run out and claim your prizes: the two Divine Blessings.]) If you drop off the side of the bridge and then turn around to face the tower, you will be facing an illusory wall. Strike it to reveal a bonfire and rest up. After you've rested and repaired your armor you can take out the remaining Bounding Demons in the field, and once they're all gone you can explore the area freely to find the two Divine Blessings hidden in the lava - if facing toward the first tower from here, you'll find them behind some roots on the left side of the field [leave the bonfire room and walk around the tower until you reach the side that is opposite of the entrance. From this point, turn and face directly away from the tower. If you run that direction, you will see two bodies behind some roots that have the items]. You can then make your way out of the lava by walking straight out from the tower bonfire and into the collapsed tower piece that's across the lava, it has a root path inside leading up to the next area.
Note: Once you get up out of the lava you can unequip the Charred Orange Ring and use something else. Although remember that if you die and respawn at the tower bonfire you'll need to re-equip it to escape the lava field again.
The Izalith Ruins
When you get to the top of root you'll enter an area comprised of crumbling ruins and columns which is populated with a wealth of Stone Demons, including some up above which can occasionally drop down behind you. Advance slowly and try not to attract too much attention to ensure you're not swarmed. You'll encounter around ten of them in a passage that goes under a bridge toward a staircase, but before you go up have a look to the left of the stairs to find another soul item. As you go up then you'll find a new enemy waiting at the top - a Chaos Eater. Take these out at range where possible as their acid attack will corrode your equipment almost immediately and their grab attack can decimate your health, if it doesn't simply kill you. Past that you'll face a junction with a choice of routes to take - one to the left with stairs, a middle path and another one to the far right with root paths leading down. The path to the left leads to the boss, so we'll take that one last.
Go to the path on the far right first and down the root to be on top of the bridge you went under earlier. The area is not very large and may still have a couple of Stone Demons in it but there's also a couple of items, one of which is located on a root over the edge on the left side. Make your way back up to the junction once you've gotten everything here and take the middle path. There's more Stone Demons down here with a cavernous passage to your left and a staircase to the right. Go up the stairs first and onto the resulting bridge
The Chaos Servant's Passage
On the bridge there is a Titanite Demon and a Crystal Lizard ( Tip: This Titanite Demon is very useful for farming as it drops two Demon Titanite once killed and also respawns.) A simple way to beat him is to use a bow to lure him in towards a vague upside-down "L-shaped" root formation on the left side of the bridge. If done correctly, he will not be able to traverse over this root and the only way he seems to be able to clear this point is by using his 'jump attack'. Continue to fire arrows at him while avoiding his lightning bolts and he'll eventually expire. Past you'll be lead into a dark passage where you'll find Solaire of Astora spouting nonsense and wearing a Sunlight Maggot on his head. The helmet has seemingly driven him crazy and turned him hostile toward you, so you'll unfortunately have to kill him. However, doing so will give you the helm he was wearing, his armour set, 2 Humanity, the Sunlight Straight Sword and a sunlight medal . ( Tip: The Sunlight Maggot won't have any ill-effect on you and will come in very useful later on in the Catacombs and Tomb of Giants) Continue down the passage to find some live Sunlight Maggots and a huge door. The sunlight maggots in the passage are harmless, so don't worry about them, and when you approach the huge door it will move out of your way and open a shortcut back to where the Demon Firesage fog gate was in the Demon Ruins. Ahead there is a root path on your left that goes all the way up to the Demon Firesage fog gate. When you reach at the top of root path look to your left,there is a soul item on the ledge. To get this reduce your equip load and and make a jump to the ledge (remember to heal before as in case of failed jump massive damage will be done which can be fatal). Now head back across the bridge and toward the cavernous passage from earlier.
Note: The above shortcut can also be opened early from the Demon Ruins side, but for this to happen you need to be a level 2 Chaos Servant, which would cost you 30 humanity. Doing so has it's advantages though, as it will allow you to skip the Demon Firesage, Centipede Demon and the whole lava section of Lost Izalith. Not only that but it also gives you the opportunity to save Solaire of Astora from the above fate so that he can be summoned for the final boss fight.
It is possible to do so without joining the Chaos Servant Covenant, however. Doing so requires a Pyromancy Flame and the Poison Mist pyromancy, which can be found on a corpse in the Blighttown swamp. Make your way to the shortcut door, and cast Poison Mist into the center of the door. It works best if you move yourself as close to the door as possible. Cast Poison Mist until you infect a mob on the other side. This mob is none other than the red eyed Chaos Bug that will drop the Sunlight Maggot. Once the Chaos Bug dies, you can continue on your way through to the Firesage Demon and the rest of Lost Izalith. If you find Solaire near the bonfire after killing the Centipede Demon, he will again appear near the Lost Izalith side of the shortcut door. If you killed the Sunlight Maggot before reaching this point, he will become summonable just before the Gwyn fight. A video tutorial on how to do this has been provided above.
The Chaos Eater Pit
On approaching the cavernous passage you'll see a corpse with an item inside. But beware that as you enter, the floor will collapse and fall away. If you hug the left wall as you enter you should land on a root rather than dropping in the pit, which is filled with Chaos Eaters. Walk down the root to enter the pit safely and you'll see a Chaos Eater on your way down on an out of reach platform - use a bow or spells to kill it - and a large hole in the floor. Now, if you have correctly followed Siegmeyer of Catarina's quest-line up to this point then you will find him standing beside the hole. Don't talk to him yet because if you talk to him enough he'll jump in and attampt to take on the five Chaos Eaters down there and likely kill himself in the process. So take out all but one or two using ranged attacks before talking to him so he has a fair chance of survival, and you can also jump down with him and help with the fight to ensure he lives if you want. (Please see here for more information on this point in Siegmeyer's quest-line). Be warned that the pit is covered in Blighttown-esque swamp water which slows movement and poisons you, so poison cures and the Rusted Iron Ring are advised to help getting around down here. Skirt around the large open shafts down here to find passageways beyond them - one of which dead ends at a chest holding a Red Titanite Slab. Explore the rest of the area for any more items and then take one of the stairs in the corners (past the pits) back up to where you can get back on the roots to exit.
On to the Bed of Chaos
With all the above done, head back to the junction and walk toward the stairs on the left. Right as you begin to climb them a Daughter of Chaos will turn the corner up ahead and start attacking you. She wields some powerful Chaos Pyromancy spells, but shouldn't prove too difficult to take out. At the top of the stairs you'll see the fog gate leading to The Bed of Chaos on your left and more stairs going down ahead. Before going toward the fog gate, go down the stairs ahead to find a chest guarded by another Chaos Eater - the chest holds the Chaos Firewhip spell. Now go back toward the fog gate and you'll be invaded by Kirk again if you are in human form - this is the last of his invasions, and if you beat the him in the Depths and Demon Ruins already then you'll find his armour set near the Chaos Servant covenant bonfire after you kill him this time. Nothing left to do now but go through the fog gate.
Boss Fight: The Bed of Chaos
Once you kill the Bed of Chaos you'll find you're stuck in the small area that housed it's heart, with only a bonfire for company. So, use the bonfire to warp out of there and back to Firelink Shrine to continue the main quest in the Catacombs.