There's a good few series on Youtube that explore the lore of the games by looking everywhere from official interviews to item descriptions and little details in the environments and characters themselves. The lore of the game is subtle and vast, it is so subtle that a lot of it is left very much up to your own interpretation, however there seems to be a few things that people are very certain about and one of those things is the timeline of events after the first flame.
We're all familiar with the opening epilogue of the game that explains some of the lore in its most concrete form:
"In the Age of Ancients,
The world was unformed, shrouded by fog
A land of grey crags, archtrees, and everlasting dragons
But then there was Fire
And with Fire came Disparity. Heat and cold, life and death, and of course.. Light and Dark.
Then, from the Dark, They came
And found the Souls of Lords within the flame."
What we can gather from the opening is that before the flame there was an unchanging world without any disparity, a world were everything was eternal, until the flames arose and with those fires came the dark flame, the one the Pygmy kept secret from the Lords. This dark flame is suspected to be Humanity and we can see that it's required for a lot of things to do with the flames such as kindling and repelling the effects of hollowing and even completely reversing the effects. However, the first flame is fading, Lord Gwyn linked his soul to the flame to keep it burning longer, but even a soul as strong as his can only burn for so long before it is completely consumed by the flames, to try to avoid this, Gwyn asked Quelaag and her sisters to try to replicate the first flame, which backfired spectacularly and formed the Bed Of Chaos and gave rise to demons and pyromancy; a form of magic strengthened by Humanity which controls flame.
The Darksign is, as has been theorized to to be linked to Lord Gwyn's hollowing due to being able to resist his soul being consumed by the flame, but it also serves a greater picture, a plot more devious than we're led to believe. The Darksign is nothing more than a calling card, the idea of being "The Chosen Undead" is a myth, conjured up by the serpents and Lords to try to find a replacement for Gwyn's soul to keep the fires burning, to manipulate you into destroying all remnants of Gwyn's rule so that the Age Of Fire can burn a little longer, just so the whole process can start again when you eventually hollow yourself and have to be succeeded by another "Chosen Undead".
This is why there's so many other people you find on your journey trying to do the exact same thing you are, for their own reasons; Sigmeyer of Caterina and Oscar The Fateless are the most obvious examples. A purpose in life prevents you from hollowing as having a will of your own to follow a goal is what keeps Humanity strong, as described in the text for the Dark Bead and Persuers spells. You're not the only one on this quest and you probably weren't the first, you're not really "chosen" you're just the one strong enough to make it to the end, and a strong soul is what's needed to succeed Lord Gwyn, so being able to fight through everything to get that far was all part of the plan, it's all just a test to prove you're the best fuel for the flame.
So, why do you need to kill him? Simple, really, his soul is the last part of the First Flame that needs to be returned, you had already killed all the other Lords and returned their souls back to the First Flame to make it whole again. However Gwyn's soul is weak; almost completely consumed by the flames and you need to link yours to the flame to be able to keep it burning a short while longer, the Age Of Fire is ever-fading and nothing will stop the inevitable, but you can postpone it with a strong enough soul, like your own.
There is another way, though, Darkstalker Kaath and his Darkwraiths are trying to find Humanity. See, as I said before, Humanity is thought to be the dark flame that the Pygmy found and kept secret while the human race split and grew, split and grew, making the task of returning this final piece of the First Flame back to itself a very daunting task, something that the Darkwraiths have been trying to do with their abilities to steal souls and Humanity directly from any who possess it. Choose not to link the flame and you can instead choose to return the final piece, the dark flame, back to the First Flame and, perhaps, either allow it to burn forever or let the world fade into an Age of Darkness. There's no way to be sure what the grand scheme is, but, either way, all you really are in this is a pawn in a great power struggle between two serpents. Choosing this ending, however means that you are returning all Humanity back to the flame means that you will have to give yours up, too, so what's really left when everything is over and what does this mean for the future?
The sad thing about the ending of Dark Souls is that it doesn't matter what you choose, the flame dies anyway; whether it be sooner or later. However that doesn't mean much, it's how you choose to have your legend live on in the new age that counts.