A Master of the Mystic Arts who lost somebody, went looking for a way to stop time taking people, and found something that offered it.
Kaecilius broke from the Ancient One’s teaching after a loss, and what he wants is not power but the end of death. Dormammu offers eternal life, and Kaecilius and his zealots take the offer without reading what eternal means in the Dark Dimension.
Mads Mikkelsen plays him with more grief than menace, which makes the ending land: he gets what he was promised.
Minor spoiler
When Strange defeats Dormammu with the time loop, the Dark Dimension pulls back and takes Kaecilius and his followers with it. They become Mindless Ones. He does live forever.