Mordred of Camelot is so Obviously Evil he might as well have a Dick Dastardly mustache. Not only is he openly proud of being the bastard (in more ways than one) son of Arthur and Morgana, he actually has a Villain Song where he outright rejects all of the virtues Arthur lives by. To be fair, Lancelot isn't always the most sympathetic character, being a blatant Canon Sue, so having the villain wear a "Hello my name is Asshole" badge may be called for.
In Pokémon Live!, the entirety of Team Rocket is this, including Giovanni.
Junior in Starship is a third generation villain and proud of it, even having a line in his Villain Song claiming that he's "evil and bad to the bone!" His rampage against the bugs is entirely For the Evulz.
Most famously, Richard III, who notes "I am determined to prove a villain." To be sure, the root meaning of "villain" is "villein, serf" — and just as "noble" or "gentleman" became a term of praise, so it became an insult, starting with meaning unchivalrous and crude. In Shakespeare's time, it still carried some of the connotations. So it could be interpreted as, "If everyone thinks I'm just a common thug, by golly I'll show them how scary a thug I can be". Interpret that thought how you will. (C. S. Lewis discusses the term at length in Studies In Words.)
Don John in Much Ado About Nothing, counseled to act nice to remain in the Prince's good books, retorts that he's always believed in being true to your nature. Since his nature is that of a villain, he's going to go ahead and be one, rather than pretend not to be — even if it's counter to his best interests.
And, of course, in case you have any doubts about that true nature, he's also a bastard. Being born out of wedlock makes you inherently a jerk.
The Elizabethans would have regarded an illegitimate-born person as a stereotypical villain. But in this case it's also justified. Don John is jealous because he will never have the status and wealth of his brother Don Pedro - because unlike his brother, Don John was born outside of wedlock.
Iago continually changes his reasoning for his actions in Othello. Originally, it was out of jealousy because he was passed over for promotion; however, he eventually abandons not only this justification, but all possible rationalizations. He is identified simply as a "Villaine" in the list of characters in the first folio, and Samuel Taylor Coleridge famously noted his "motiveless malignity". Iago himself, when questioned on his motivation, simply refuses to explain anything...though there are those who attribute his actions to less incomprehensible motives.
Or there's the possibility that he's simply jealous of anyone who has anything good that he doesn't and fucks with Cassio, Desdemona, and Othello accordingly.
Then again, in the play he says it is "thought abroad that twixt my sheets/He's done my office," when referring to Othello. In other words, it's rumored Othello and Emilia are themselves making the beast with two backs.
Aaron in Titus Andronicus states outright that all he wants to do with his life is be evil; the only reason he aids the play's antagonist instead of working against her is because it gives him a chance to do very evil, uncouth things. It's because he's black, and black people are evil. His last line in the play is:
Aaron: If one good deed in all my life I did, I do repent it from my very soul.