I’ve read (from official sources) that the idea is that Snoke told Kylo that Vader’s redemption was actually a moment of weakness, that he was seduced to the light side which lead to his death and the destruction of the Empire (much the same way Luke was told Anakin was seduced to the dark side which lead to his “death” and the destruction of the Republic). Kylo’s fears of turning back to the light echo Luke’s fears of turning to the dark.

Anyway, when Kylo says he’ll finish what Vader started, he means what Vader didn’t get to finish because of his “redemption” (and the what being I assume the eradication of the Jedi - so Luke, who from his point of view is analogous to the Emperor from Luke’s).

A reall fascinating twist on the established story, I think.