Dark Souls

To be able to even ring the bell you need to find Quelaag's Domain.

Preparation

Firstly, starting from the bonfire in the tunnel (the second bonfire in Blighttown), you'll want to become human. I also highly recommend that you kindle the fire.

Why? because a few steps away from the bonfire, a NPC Phantom will attack you.
She is pretty easy to kill, you shouldn't have a problem, just keep your shield up and attack after she's taken a swing.

After killing her, you get her meat cleaver, BUT MORE IMPORTANTLY, you will now be able to summon her to assist you in the battle against Quelaag.

Finding Quelaag

Now, actually getting to Quelaag. She isn't too far away, but the issue is getting across the swamp.
Do you see that white lump in the distance? That's where she is.
Sure, you can go all the way round taking 10 minutes and using all your Estus after the golems crush you repeatedly.
OR you can dash across, sticking to the right, only facing 3 big bugs.
Sure, you'll get poisoned. But in the grand scheme of things, this will have very little effect. It'll run out before too long.

When you get to the beach area, and enter the little cavern, abstain from attacking what appears to be 2 guys with some nasty-ass eggs on their backs. They will not attack, so long as you don't.

You will see the summon sign for the NPC you defeated earlier. Her purpose will be to divert Quelaag's attention and attacks from you, helping considerably.

Attacks (courtesy of the Dark Souls Wiki)

Slashes: She slashes either left to right or right to left. Shield or roll to avoid these. Not a huge threat.
Stab: This attack has much more range that you expect it to. Can be a significant threat.
Lava: When she pats the spiders head or the spider looks like he is going to vomit she will use this attack. She will spit lava on the ground around her, if you are far enough away she may shoot 4-5 balls of lava at you.
AoE Explosion: Whenever she reaches down the sides of the spiders head she is prepping her AoE attack. Back up keeping your shield toward her. Large Threat!!

Strategy

The core strategy for fighting Quelaag is to stay to her front sides, waiting for her to spew lava so that you have time to do some damage.
She dramatically telegraphs her movies so after a few attempts there are not many moves that will surprise you.
Use the Black Knight Shield, or some other high fire resist shield, and fire resistance/physical resistance armor as this helps stop much of the damage from her fire AoE attack.

Rewards!

You will recieve the Soul of Quelaag. Don't use it, as with the help of a blacksmith you can re-create her sword with either a Katana or a Scimitar!
oh, you also get a nice 20,000 souls too.

Once you've done with her, you simply go up the stairs, pull the lever and net yourself 15 gamerscore!

*If you downvote this, please tell me why in the comments so that I can improve it*

CrackforBrumak's solution is spot on, but in regards to strats you really do not have to do much if Mildred is there.

The Maneater known as Mildred will take 99% of Quelaag's attention - all I did was hang back and shoot her with arrows. Mildred is a tank, so most of Quelaag's attacks, although deadly to you, on your first playthrough will be nothing to your friendly NPC.

Let Mildred take the lead, and just keep your distance. If you have magic, either from the sorcerer tree or the miracle tree, then you can get this done faster. Pryomancy does not work as well (nor should it - she is fire! ), and arrows will take a bit longer, but they still work.