The narrative voice in Notes from the Underground is, of course, that of the Underground Man. And in fact, much of the text is just his voice, talking to us.
Since you could imagine this getting a bit uneventful, Dostoevsky does something interesting: he creates a second character, an imagined reader to whom the Underground Man is speaking. The narrator imagines our responses to his argument, which means his voice takes the form of a retort. But don't take our word for it: "the antithesis of the normal man […], the man of acute consciousness, […] has come […] not out of the lap of nature but out of a retort, [a] retort-made man." So that's your narrative voice: a retort.
When the Underground Man later declares that his readers don't exist – that these "gentlemen" to whom he has been speaking are imagined – he renders absurd his entire "retort." He's responding to nothing. He's a retort to a voice that doesn't exist. And that's quite a premise for a first-person novel.