Dostoyevsky’s Notes From Underground, although “villain” might be a bit strong to describe the character.
Iain Banks’ The Wasp Factory
John Fowles’ The Collector starts and ends from the villains point of view, but has a middle section from the point of view of the victim.
An excellent question. In the cases of the “my side” novels, like the Dracula and King Kong ones I list above, the character is clearly the villain in the original work.
American Psycho by Bret Easton Ellis, although Bateman’s more an anti-hero.
I know it’s not a book but I wonder if Dr. Horrible’s Sing-Along Blog would qualify?
If we’re including anti-heroes, Andrew Vachss’s Burke series is told in first person by Burke, a very anti-hero character.
“For Love of Evil”, the sixth book in Piers Anthony’s Incarnations of Immortality series.
Jean Rhys wrote the Wide Sargasso Sea which is a “prequel” to Jane Eyre from Rochester’s wife’s perspective.
There’s always Jeff Lindsay’s Dexter series. The TV series was inspired by (heck, the first season absolutely paralleled) the first book, but the books and TV series seriously parted ways immediately thereafter. Dexter occasionally breaks some laws to do what he feels is the right thing. You know, laws like the one against cold-blooded murder.
By the way, you might want to pretend that the third book in the series doesn’t exist. It’s a major misfire.
I never read the book and it’s been a long time since I’ve seen the movie and I assume this isn’t quite what the OP had in mind since it’s not like we’re going and following the villain as he commits his crimes, but how about Dead Man Walking.
If that counts, I’m culturally required to point both Lazarillo and El Buscón. This Penguin edition gets you both together!
I was going to mention El Lazarillo! 
At least he occasionally gets punished, though.
Peter Watts short story “The Things” is a retelling of the ill fated arctic outpost from the monsters perspective.
I think it’s in his newest collection, “Beyond the Rift”
How about Wicked?
They’re certainly very naughty boys, but I don’t know as I’d call them “the villain.” It’s all a matter of perspective, and from theirs (and thus, the reader’s), they’re anything but. The Grey King is the villain in the first book; I’m sure they’ve got others in the latter books.
Technically, that also fits the aforementioned Flashman; he’s the bully who antagonizes our hero in Tom Brown’s School Days before getting expelled for drunkenness, and the idea is that he’s now narrating his adventures for us – as an unrepentant coward who rapes and cheats his way through the novel while lacking even the decency to pay up once the offer of a bribe has already gotten the job done.
And the same author wrote “Confessions of an Ugly Stepsister”
Jesse Bullington’s The Sad Tale of the Brothers Grossbart has the titular brothers steal and murder their way across medieval Europe and the Middle East, always just a few steps ahead of a literal pack of demons called up in revenge by the survivors of their vile enterprises.
I don’t know, don’t you think Elphaba comes off too sympathetic to be a true villain in this story?
A while since I read them but the Bernie Rhodenbarr (Burglar series) books by Lawrence Block.
In the superhero deconstruction From the Notebooks of Dr. Brain the narrator is strongly implied to be an escaped supervillian with mind control powers.
*Banewreaker *and Godslayer by Jacqueline Carey. In my opinion, it’s basically LOTR told by the bad guy’s POV.