People have been talking about the villain's POV in other responses so I thought I'd look at the other angle: where we have no direct insight to the villain, and just see them from the outside.
At some level, there must be a logic behind a character's action. Even if a character behaves differently on different occasions, well that unpredictability is evidently part of their logic, and all the more frightening to do so.
People like to fill in the gaps in their understanding of others, and the same is true of readers' attitudes to characters. If you can set up clues that relate to an underlying motivation for the character, then the readers can put the pieces together and you can build a character more powerful and more frightening than you could consciously.
A traditional approach which has much merit, not because the events of the backstory will appear directly or indirectly in the story, but because this will make the villain more real in your mind, and there will be more pieces that suggest themselves to you, to bring into the story.
Another traditional idea which has a lot of merit is that of introducing a chink of light into the darkness: this is something that will give the reader's thinking more to chew over. Curiously, this doesn't break the logic, but makes it more real.
One of the most frightening villains for me is Bill Sikes from Oliver Twist. We have no insight into the internal world of this appalling character, and yet even he arguably has a moment of neutrality, as he picks up Oliver and carries him off after the failed burglary.
Another very convincing villain for me (who I won't name to avoid spoiling) is the killer in "Sleeping Murder" by Agatha Christie. Almost by definition in a detective story we don't have access to the interior world of the killer, but here is someone who to me is totally convincing in their black-and-grey shadow, and much of the power of that novel (one of Christie's best) comes from this appalling character.
I want to end with yet another trope. In art, a character is suggested with some squiggles of paint, and it's no lie to say that the act of creation is as much in the reader's mind as in the writer's. When summoning a villain, this can be the more powerful if somehow the writer can tap into the fears and hatreds that lurk in all of us. So the villain is the more horrible for being part of ourselves.
Enough. Have fun.