"Greetings, Undead warrior. I am the primordial serpent Darkstalker Kaathe. I can guide thee, and illuminate the truth. Undead Warrior, conqueror of the Four Kings; is this not your wish? To know the truth of men, and the Undead?"
Darkstalker Kaathe greets the Chosen Undead and tells his side of the story: that the player is not to renew the Age of Fire by rekindling the First Flame, but is in fact the descendant of the one who possessed the "Dark Soul", and shall be responsible for bringing about the Age of Dark. Kaathe will transport the player to Firelink Altar in order to place the Lordvessel, and then instructs the player to collect the other Lord Souls in order to confront Gwyn, Lord of Cinder.
Placing the Lordvessel while under the care of either of the serpents will result in the loss of the other. Frampt provides the service of breaking down larger Titanite pieces and can get rid of items in exchange for souls, while Kaathe provides Cracked Red Eye Orbs to the player while a member of his covenant.
It is possible to join the DarkwraithCovenant without placing the Lordvessel and thus not alienate Kingseeker Frampt. After answering "Yes" to Darkstalker Kaathe's question and being transported to the Firelink Altar - do not place the Lordvessel. Instead, use a Homeward Bone or the Homeward miracle to exit. As long as one does not place the Lordvessel, neither serpent will depart. In this way, the player can be both part of the Darkwraith Covenant and use Frampt's "Feeding" service.
Kaathe will depart if the Chosen Undead places the Lordvessel while Kaathe is not present.
In Dark Souls III, when the player kills Yuria, she says "Kaathe, I have failed thee", proving that she and the other Sisters of the Sable Church, and thus Londor by extension, are followers of Kaathe. In the original Japanese she says "...Kaathe... your dying wish was not fulfilled..." which would imply that despite having followers, Kaathe is no longer alive during the events of Dark Souls III.
Greetings, Undead warrior. I am the primordial serpent Darkstalker Kaathe. I can guide thee, and illuminate the truth. Undead Warrior, conqueror of the Four Kings; is this not your wish? To know the truth of men, and the Undead?
A wise choice, Undead warrior. Prove you must, that the truth becomes you. Seek Anor Londo. And claim Gwyn's heirloom, the Lordvessel. Now go, Undead warrior. Show the world that the truth becomes you. (Note: This dialogue may be different if the player has already acquired the Lordvessel.)
...Very well. Warriors as yourself are few. I will be patient. But I cannot wait forever...
After answering "No"
What is it? Perhaps you have changed your mind? (The Answering "Yes" dialogue will now be said.)
After acquiring the Lordvessel
Hmm... You are astonishing. The truth I shall share, without sentiment. After the advent of Fire, the ancient Lords found the three souls. But your progenitor found a fourth, unique soul. The Dark Soul. Your ancestor claimed the Dark Soul, and waited for Fire to subside. And soon, the flames did fade, and only Dark remained. Thus began the age of men, the Age of Dark. However... Lord Gwyn trembled at the Dark. Clinging to his Age of Fire, and in dire fear of humans, and the Dark Lord who would one day be born amongst them, Lord Gwyn resisted the course of nature. By sacrificing himself to link the Fire, and commanding his children to shepherd the humans, Gwyn has blurred your past, to prevent the birth of the Dark Lord. I am the primordial serpent. I seek to right the wrongs of the past, to discover our true Lord. But the other serpent, Frampt, lost his sense, and befriended Lord Gwyn. Undead Warrior, we stand at a crossroads. Only I know the truth about your fate. You must destroy the fading Lord Gwyn, who has coddled Fire and resisted nature, and become the Fourth Lord, so that you may usher in an Age of Dark.
Very well! I shall now guide you to Gwyn's prison. Be still. Entrust thine flesh to me. *Kaathe will take the player to Firelink Altar.* This is Gwyn's prison. Now, place the Lordvessel upon the altar.
Fool... You could not be the Dark Lord. Enough of this...and farewell to you. I shall return to the Abyss, and await the true Lord of Men.
After placing the Lordvessel
Very well. Once the vessel is filled with souls, the gate to Gwyn shall open. Seek Gravelord Nito, the Witch of Izalith, and the traitor Seath the Scaleless. Fill this vessel with their souls. Then, the gate will open... so that you may kill Gwyn. ... Are you ready? Then, let us return to the Abyss. Entrust thine flesh to me.
What is it? I am your guardian. Go on, state your wish.
Darkwraith Covenant Entry
Ahh... If you wish, I shall grant the art of Lifedrain, the legendary power of the Dark Lord. It can preserve your humanity while Undead, and cast off the shackles placed upon your brethren.
Talking before acquiring the Lordvessel
Undead warrior. To speak now is premature. It begins with your retrieval of the Lordvessel.
Talk x1 after placing the Lordvessel
They failed me, every last one of them. They were strong, but saw not the truth... I am certain that you will prove different.
It's actually possible to kill Darkstalker Kaathe, but only through the use of third party programs. Kaathe has 24,464 HP and drops 5,000 souls upon death.