Darkstalker Kaathe is a Primordial Serpent, like Kingseeker Frampt.
Kaathe greets the protagonist 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 in bringing about the Age of Dark. Kaathe will transport the player to the Kiln of the First Flame 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.
Kaathe's intentions are very different from those of Frampt. He's the leader of the Darkwraiths and wants to see the Furtive Pygmy's plan come to fruition by recruiting Undead with the Darksign, so that they can become the new Dark Lord, heralding the Age of Dark. It's implied that he's the one who manipulated the people of Oolacile to awaken Manus, and offering the art of Lifedrain to the Kings of New Londo, indirectly responsible for the desctruction of Oolacile and New Londo.
If the Chosen Undead sides with Kingseeker Frampt by placing the Lordvessel for him, Kaathe won't appear after the Four Kings are defeated. If the Undead already met Kaathe, the next time Frampt is spoken to, he will scold the Undead and leave permanently.