Keep in mind I’ve only seen the first season, but villain Prince Zuko from the television series “Avatar” genuinely loves his uncle. He chooses saving his kidnapped uncle over his driving obsession to find the Avatar.
Oh please. Ivy’s meaner to her than the Joker is.
I first saw HQ in an ep of Batman:The Animated Series, in which it was clear that the Joker was abusive to her. There was even a scene of them at home in which he was wearing boxer shorts and a WIFE-BEATER T-SHIRT (and I hadn’t heard the term at that time). IIRC, she ended up in the hospital & Batman was trying to convince her that she didn’t have to put up with the abuse.
Now I see on this page that HQ is basically a dingy blond screw-up, which makes me wonder-
SHE was a psychiatrist at Arkham Asylum?!?!
To an extent. She certainly isn’t stupid, but she’s known to have used seduction to get an A in at least one of her classes, so what she lacks in pure smarts, she makes up for in ambition and ruthlessness.
Oh, and as for the OP, there’s an episode of Law and Order: Criminal Intent featuring a longtime con man. He’s defrauded thousands of dollars from vulnerable people, with the (unwitting) help of his girlfriend, an epileptic who thinks that her sensory hallucinations are visions. He has a client’s son murdered to trick the client into believing that she has psychic abilities, in order to gain a huge payday. But he genuinely loves his girlfriend; he’s even spent a lot more time in prison than he’s needed to in the past to keep her free.
Goren and Eames use his love against him, threatening to tell his girlfriend that he knows that she isn’t psychic unless he confesses to the murder and implicates the powerful criminal he got to commit the actual deed.
You must be kidding. I agree that Ivy is abusive to Harl (who clearly exudes some sort of weird “abuse and sexually exploit me” pheremone – but on the inevitable day that Harley’s decayed, half-eaten, alligator-raped body is found floating in the Gotham reservoir, Mr. J is going to be the murderer and Dr. Ivy will be the avenger.
And then Batman will intervene to save the Joker, as Bruce is an idiot.
Dr Claw and his cat 
Any old AD&D Ravenloft fans? Strahd von Zarovich’s unrequited love for his brother’s fiancee, Tatyana, is one of the best stories IMHO.
But he’ll have killed her because she makes him love her. I’ve got a friend who analyzes their relationship
The dragon fell in love with Donkey in Shrek.
Babylon 5: Alfred Bester and his “blip” girlfriend from “Ship of Tears”.
I think he’s attracted to her, but ultimately turns on her brutally. I don’t think Goeth was capable of loving anyone.
Cersei loved Jaime in the sense that loving him was a manifestation of loving herself – she’s an extreme narcissist. He’s her twin and worships her and looks like a male version of her, so being with him is the closest she can come to being with herself. All the other men she’s with are either Jaime substitutes (Lancel) or guys she bangs because she thinks they’ll protect her or get her somewhere (the Kettleblacks, etc.) Her love for Joffrey is blinding, but her younger children are mostly pawns for her own political ambition, although she does at least have some maternal instincts towards them.
Not really a villain but does some bad things: Roy Batty (Pris).
To continue in this vein (heh), I would assert that Lilah Morgan (of Wolfram & Hart) loved Wesley about as much as she was capable of loving anyone.
Bester doesn’t just love his girlfriend, though - he’s a telepath “nationalist”, and he cares deeply for his fellow teeps.
Um, who are Cersei and Jaime? That is, what story do they appear in?
I don’t think Spike qualifies under the “one and only one person” stipulation of the OP. Spike also loved Buffy, Joyce, and Fred.
And Dawn. And his mother. And Angelus. (But not Angel…he loathes Angel.)
Since Harley Quinn was mentioned she is an example of this for Poison Ivy, as well - she’s the only human Ivy can even stand, let alone care about.
I skipped Dawn deliberately because of some things that happened in the fifth season, but thinking about the sixth, I’d say that I was wrong; he did love her. And, hell, he was quite fond of Willow–not fond enough to spare her life, of course, but enough to give her the option of being turned rather than outright slaughtered, and enough to reassure her that she was attractive when she began to put herself down.
That would be George R. R. Martin’s A Song of Ice and Fire series (starting with A Game of Thrones), as noted by hdc_bst (post #57).
One of my own favourite bits from an altogether tragically truncated series. Brimstone was approximately one trillion times better than all the Ghost Whisperer clones combined.
Anyway, does Ernst Blofeld count? He sure loved that white fluffy cat.