Soooo, I'm trying to rewrite it here. I've tried to avoid Forwood bias and speak from just Caroline's POV as if she isn't in love with Tyler in canon. Because it's not about whether Tyler is better than Klaus: it's about whether Klaus is good for Caroline.
1. Klaus gives her gifts and compliments, he's chasing her.
It's flattering, especially since Caroline has always been the girl who "tr[ies] so hard and [is] never The One". Look at her in season 1 with Damon, in season 2 with Matt. Even in her friendships she is always second- see how Bonnie and Elena treat her in the first two seasons. So when Klaus, an Original- "the most powerful creature on the planet" no less, chooses HER people expect her to play blushing maiden and accept.
She's won! She's getting diamonds and ball gowns and champagne! He's pursuing her so ardently! Why is she refusing him? She should at least give him a chance.
See, in media this is often how romance is framed. Witness Bond girls, Scarlett O'Hara, etc. steadfastly refusing a man's advances until he finally forces a kiss on her and... she realizes she was in love with him all along!!! *they are swept away by passion*
The thing is: the girl is not allowed to refuse. She's just playing hard to get; she doesn't know her own mind; she's too shortsighted to see his charms. It's based on the stereotype that a woman doesn't know what she wants and therefore a man should decide. Need I point out how wrong this is? But it's a well-established idea, so a lot of people have internalized it.
Caroline knows her stance- she realizes that he is trying to "buy [her] off", while he's telling her to "mark [his] words" that her choices "won't be enough for [her]". She's told Klaus multiple times to leave her alone, that she chooses Tyler- he just doesn't respect her opinion.
2. Klaus has promised to show her the world, "art, music and great cities".
There's that "freaky Cinderella fetish" of having an older, experienced man sweep you away to the life of your dreams. You get a world tour, first class hotels and gorgeous clothes to doll you up. And he prizes you, VALUES you.
Which might be OK if he weren't objectifying you, treating you like a prized pet instead of a person. In 3x15 Klaus makes it clear to Kol that he is pursuing Caroline not because he thinks she's a wonderful person, but because she's a "challenge". There's also the sadistic thrill of taking something from Tyler, of asserting his "alpha male" status. Caroline is an object to him the same way his hybrids, his paintings, Elena and that favorite horse of his are: precious due to the difficulty in attaining them but not people with feelings and rights that are equal to his.
He expects his hybrids to serve him the way he has witches and Stefan serve in seasons 2 and 3- they are "mover slaves" and "bodyguards" that are not allowed to have aspirations and other loyalties. "[Their] existence is to serve [him]. To please [him]." Those that claim that Caroline would be the exception to the rule are being naive- that is just not how abusers act, even when they get their "one special girl." (See: Elena and Damon) The way he turns on her with murderous fury in 4x01 and 4x06 should make that clear- "Caroline you're beautiful. But if you don't stop talking I will kill you." And from the episode blurb for 4x13, I'm pretty sure we will see Klaus strike her at last.
ETA : Proving my point exactly: he skewered her, drank from her (just like Mikael did with other vampires, which emphasizes those abuse cycle flags) and since he is a hybrid, left her with werewolf venom in her system. ALL DELIBERATE. There is no handwaving this away- this is textbook abuse.
Caroline knows this too, that he "collect[s] hybrids" and "can't connect with people because he doesn't even try to understand them." And puh-lease. She can show herself the freaking world without being some man's arm candy. The very idea that you need a man to show you how and where to travel gives this avid traveller actual RAGE. You get a passport, you book a ticket and you get on a plane. NONE OF THESE STEPS REQUIRES A MAN. As if Klaus would drop his agenda to even accompany her on a world tour anyway- what are you smoking, people?
3. Klaus has saved her life.
This is usually a precursor to romance in a lot of media/stories. Hell, I think it is awesome that Tyler runs to save her in 4x08 and 4x10- love is a great motivator, after all. But look at the facts:
Klaus endangered her life in the first place by forcing Tyler to bite her in 3x11, to force Liz to back him staying in MF to the Council, and get back at Elena for defying him. He dismisses her suffering as "collateral damage".
He took credit for ELENA saving her life in 3x21. He grabbed her in the hallway and gave her a speech about how heroic he was as Caroline was running. That's not saving her life. That's what Elena was doing, holding the knife to her own throat.
He pervs on her for services rendered. 3x11 is the worst- he forces her to drink from him on her birthday and pretends it's a gift. No, he doesn't have to- he gave Damon (2x22) and Stefan (3x02) the cure via bottle. He only does this to female vampires because according to Julie Plec, bloodsharing for vampires is "like naked cuddling". THAT IS ASSAULT and just out and out gross. And look at the shame in her expression as she drinks.
ETA: After a couple K/C comments, I think I have to make the context of this scene very plain, because it is not "heartwarming" or romantic in the slightest, even though the show has framed it that way. In plain terms: A serial killer, (who has tried to kill you, your boyfriend and your best friend, killed her aunt, turned your boyfriend into a vampire, bodyjacked your teacher, and mindjacked your mento and your boyfriend) forces your boyfriend to inject you with poison. You have painful hallucinations and are on the verge of death. He has the only cure. Instead of handing over the cure, he forces your mother to admit him into your home. You don't want to die- it's your birthday. But wait! There's a catch- while he gives you the cure you have to naked cuddle with him on your bed. So even though you don't want him there, you dislike him, are frightened of him and you don't want to do this, you're forced to naked cuddle with him to stay alive.
Horrifying when you strip off the soft lighting, sweeping background music and promises of "great cities, and art and music", isn't it? (The only other scene more squicky is when Elena drinks from Klaus in 4x03 in front of Stefan. Stefan, the guy who punched out Damon for having Elena drink from him in the previous episode, begs for the cure an is forced to watch while the love of his life does the exact same thing.)
In 4x01 Klaus does try to correct her but he's hardly pushing her off or confessing he's bodyjacked her boyfriend as she's frantically kissing him and taking their clothes off. She's the one who aborts sex when she realizes he isn't Tyler. ETA: Astxgirl0302 states in the comments: this is also sexual assault.
All in all- it's not noble or romantic when he "saves" her. Even Caroline dismisses his rescue in 4x01, where he doesn't have any previous agenda other than saving her, as "semi-decent".
ETA: I see the Klaroliners applauding the scene at the end of 4x13 for the same reasons and am just horrified. This is like some guy raping you and then handing you an ice pack to relieve the swelling. NOTHING IS TAKING AWAY THE EMOTIONAL TRAUMA OF AN ASSAULT AND THE PAIN SHE ENDURED. Don't even get me started on how he is stroking her as she drinks to save her life.
4. He is a hot woobie bad boy (See: Damon Salvatore) with bonus cute accent. He's badass, worldly, rich and powerful. He knew Dickens, has been hiking in the Andes. He's admitted how sad and lonely he is. After all, his own parents hated him- one abused and then hunted him for a thousand years and the other put a curse on him (and also tried to kill him). He was also tormented by the hunter's curse for 52 years (4 months and 9 days). Oh, and he's an artist- see him paint sensitive post-modern snowflakes for charity. Basically: he's attractive and you feel sorry for him. You want to give him a hug.
But see how he treats the people who love him? (See: oft-daggered siblings and murdered mother) That is abusive behavior, not just a sign of his inner pain. He doesn't have to be lonely- he chooses to be by hurting those who love him. Rebekah, especially has stood by him through the years. And yet he dismisses her steadfast love by choosing to create hybrids instead, by disparaging and daggering her when she dares to have a different opinion.
So many Klaroliners claim Caroline should use her caring to save him- it changed Tyler. And I say: bullshit. Because that is not how it works: abusers continue to abuse until they choose to break the cycle no matter how much love their victims give. (Again: see Rebekah.) Tyler himself wanted to change- he admits as much in 2x01 ("I don't want to be like this anymore.") before he and Caroline were even friends. He knows he's been abused, knows he has rage issues and he wants to break that cycle. He apologizes when he lashes out, takes responsibility for the things he does wrong and acts to keep himself from doing so again. Klaus on the other hand still feels entitled to deal out abuse- he blames the murders of the hybrids and Carol Lockwood on them- because his feelings got hurt, it forced him to kill them.
Caroline, quite rightly, wants no part of that. She's done it all before with Damon, remember? She doesn't want to get death threats mixed with compliments, to struggle to please somebody to prevent them from hurting her.
5. The Madonna/whore complex. "She keeps spreading her legs for Tyler, ewwww. Proper ladies prefer romance, not sex. Klaus treats her like a lady, they only flirt and go on dates. That is pure love."
I'm paraphrasing from Tumblr because this keeps coming up as an argument for Klaroline. FYI: Klaus has sexually assaulted her. And it's not wrong for Caroline to enjoy consensual sex with her boyfriend, especially since it's established in canon that they both have higher than average sex drives. If you have a problem with teenage/pre-marital sex being depicted on TV, you are absolutely watching the wrong show.
6. The directors have framed Klaroline scenes as romantic. I too hear the soaring instrumental background during his charismatic speeches and see the way they dress Caroline in sparkles and gold, how they dressed Tyler in black, Klaus in white and Caroline in red in 3x20. Almost every interaction Caroline has with Klaus is in a date-like context- drinks at a bar, sitting on a bench chatting over champagne or even full on dancing together at a ball. They wanted to have another romantic triangle in the show, and there's chemistry between the actors. So I see where Klaroliners are coming from- we're SUPPOSED to see how Klaus is interested in her, and that she is intrigued by what he offers her. But time and again she has rejected his advances, downright plotted to kill him and for good reason.
I know they're supposed to have a confrontation coming up, so we will probably see this triangle collapse soon, if the murder of Carol hadn't sunk the Klaroline ship already. But I just thought I'd write down my thoughts on this.