A somewhat mysterious, but clearly violent and unstable young man, Rogozhin wants desperately to marry Nastasya—but it clear that he will harm her in some way once he does so.
You know what would be helpful when talking about Rogozhin? If we maybe step back in time a bit, and talk about the very first completely evil character in literature. That's right, people, we're talking about Shakespeare's Iago (from Othello).
Now, for sure, before this play, there were lots and lots of fictional bad guys out there. But all those preceding bad guys had one thing that made them slightly less scary in the audience's eyes: a motive. A reason for why they did what they did—revenge, love, ambition, anything that the audience could point to as a reasonable cause and relax because the universe still made sense.
Suddenly, though, we get Iago, who does what he does for no reason that we can figure out. And what's more, at the end of the play, he refuses to explain himself, saying that he will never talk about any of it ever again, taunting, "all you know, you know." There is something about this inexplicability that makes him so much, much scarier than your run-of-the-mill killer, no? The same effect is had by our favorite sci-fi villains—the Borg from Star Trek or the planet Solaris from Solaris.
And there's a pretty sizable chunk of that Iago-style of just straight up evil without a cause in Rogozhin as well. Some critics complain that he's not a well thought-out character, but maybe that's just exactly the point. Everything about him—from his creepy, haunting eyes that can be seen at any distance by Myshkin or Nastasya; to his gloomy, depressing, horrible house; to that unnerving painting of the dead Christ that he's got hanging up in his room—all adds up to a kind of supernatural aura of evil, but also at the same time to nothing tangible.
No matter how much we know about him, we still don't know why he kills Nastasya when he finally does—and he doesn't even try to explain himself when he has the chance. Here is all he can tell Myshkin about it after Myshkin asks him, "[T]ell me—how did you—with a knife?—That same one?":
"Yes, that same one."
"Wait a minute, I want to ask you something else, Parfen; all sorts of things; but tell me first, did you intend to kill her before my wedding, at the church door, with your knife?"
"I don't know whether I did or not," said Rogozhin, drily, seeming to be a little astonished at the question, and not quite taking it in.
"Did you never take your knife to Pavlofsk with you?" "No. As to the knife," he added, "this is all I can tell you about it." He was silent for a moment, and then said, "I took it out of the locked drawer this morning about three, for it was in the early morning all this—happened. It has been inside the book ever since—and—and—this is what is such a marvel to me, the knife only went in a couple of inches at most, just under her left breast, and there wasn't more than half a tablespoonful of blood altogether, not more." (4.11)
What is so creepy and striking is that total lack of emotion—just cold and rational description of the stabbing. Rogozhin so embodies that evil impulse that he can't even put together the timeline enough to remember whether he'd been planning to kill her all along or not. But at the same time, look at the level of detailed description we get about the placement of the knife, all the with same tone, whether he's talking about hiding it in a book or in Nastasya's chest.