Tragedy
We'd be hard pressed to call this anything other than a straight-up tragedy, right, folks? The extremely depressing ending is really pretty littered with bodies, both literally and figuratively speaking. We've got the actually dead (Nastasya, Ippolit, and General Ivolgin), we have the functionally dead (Myshkin and Rogozhin), and we have the might-as-well-be dead (Aglaya).
And just like in the best of tragedies, most of them have been brought down by some fatal aspect of their character that just cannot get along with the rest of the world: Nastasya comes with a built-in death wish, General Ivolgin basically drinks himself into an early grave, Myshkin is too good and pure to function as an actual human being, Rogozhin can't really tell the difference between his own love and hate, and Aglaya is so in love with the idea of being with an idealist that she can't see past the end of her nose. No one gets what he or she wants, and everyone's plans for the future are ruined—so, yeah, tragedy all around, y'all.