Count Alexis (or Alexei) Vronsky ain't a particularly rounded character. We know that he is handsome, charming, and an officer in a regiment of soldiers. But beyond that, his function in Anna Karenina is to be the man with whom Anna Karenina has an affair and a child, and eventually flees from in suicide. He's the object to her subject; the butter to her toast.
Even though Vronsky sometimes seems to be a plot device wrapped in the shell of a good-looking soldier, Tolstoy does give us some insight into his psychology. What we want to know is: how does this up-and-coming young officer get involved in a soul-destroying affair with a married woman? Here are a few ideas...
In Levin's "Character Analysis," we talk about how Levin's marriage keeps him on the path to righteousness. Levin is sentimental about family life and the family unit as the ideal mode of social organization to keep all of its members healthy and happy. And Vronsky is Tolstoy's example of what can happen to people who don't value family life the way Tolstoy thinks it should be valued.
Vronsky has never had much in the way of family life. His father was a nonentity Vronsky barely remembers, and his mother appears to have slept around a lot. Eventually, Vronsky was sent to a boarding school, where he learned great manners but not much about morality. (Remember that Tolstoy is reeeeeally suspicious of the effect of education on the human heart and ethical sense.) Vronsky emerges from his childhood without any intention of getting married and starting a family. He doesn't see the value of marriage when he can sleep with any woman wants, and he also thinks husbands are ridiculous, pointless figures to be laughed at while you seduce their wives.
Vronsky believes that human relationships should be a matter of pleasure rather than a moral issue. Because Vronsky views marriage as a ridiculous social institution, he has no means with which to regulate his unhealthy relationship with Anna. He goes ahead and pursues her, even though she initially resists his seduction on moral grounds. When she finally sleeps with him, he realizes only then that he may have done something terrible to the woman he loves. By consummating their affair, Vronsky has changed Anna, irrevocably, from a morally upstanding woman to a morally reproachable one. Yikes.
After this, there's the ongoing question of whether they should get married or not, although the point seems to be moot. Sure divorce and remarriage might give Anna some social standing, but Vronsky's relationship has already totally altered Anna's nature.
Remember how Vronsky views husbands as pathetic? Well, that comes back to haunt him when he meets Karenin at what Vronsky thinks will be Anna's deathbed, in Part 4, Chapter 18. Karenin has always seemed ridiculous to Vronsky, but Karenin's forgiveness of Anna and Vronsky seems so noble that Vronsky physically stoops in front of Karenin. Vronsky realizes then that he has not loved Anna as he should have. If his love had been genuinely selfless, he wouldn't have ruined her by convincing her to sleep with him. And it's this thought that drives Vronsky to attempt suicide (though he goes back to his old ways pretty dang quickly).
Fast forward to the end of the book, and we see that Vronsky's lack of interest for family life continues. Despite feeling that he's bound to Anna forever, he's cold and resentful towards her.
What's more, Countess Vronsky, Vronsky's mom, still has a bad influence on him. As an adulteress herself, she's what Anna might have been with Seryozha if Anna hadn't decided to take her own life. Countess Vronsky is incredibly ambitious on her son's behalf, and hates Anna's guts. Interestingly, the Countess doesn't care that Vronsky seduced a married woman, but she takes issue with the fact that her son is a little too in love with Ana. She doesn't like that her son is sacrificing his brilliant military career to hang out with Anna in the countryside. Countess Vronsky is another example of the kind of selfish, cold society woman whom Tolstoy seems to detest (see also the arch-hypocrite, Countess Lydia).
Another thing that's troubling about Vronsky's family life is his lack of interaction with little Annie, his daughter. The narrator tells us explicitly that Karenin likes little Annie, even though she's not his, but we never hear a peep about what Vronsky thinks of her. In fact, after Anna's suicide, Vronsky gives up Annie to Karenin, to be raised with Seryozha. Vronsky's mom tells Levin that Vronsky's sad about losing Annie—but what does his mom know? We don't really have any proof of how Vronsky actually feels about his daughter. He might not care at all about Annie.
Without the guidance of an ideal of family life (such as Levin has), look how Vronsky winds up. He's got a daughter he doesn't get to visit, a mother who's eager to see him shipped off to war (where he wants to be killed), and a partner he loved and destroyed. This is all super-duper depressing.
As a cousin to Princess Betsy Tverskoy, the queen of Petersburg's naughtier high society, it makes sense that Vronsky's views would be as immoral as hers. In Part 1, Chapter 34, the narrator makes it very clear that Vronsky thinks that 1) people who get married in churches, who are modest and hard-working and honest—are ridiculous; and 2) the best kind of people are handsome, fun-loving, and carefree. Neither the women nor the men of his acquaintance are too weighed down by their consciences. They do what they want, when they want.
Then Vronsky meets Anna, and this upsets all of his carefully devised social rules about how to have affairs elegantly and easily. Vronsky discovers that forgetting about morals to live for pleasure only works as long as you're having fun. But Vronsky's affair with Anna isn't fun exactly. What he feels for her is so strong that it makes him forget his ambitions in the army, his social duties, everything. He just wants to be with her, and he sees how much the secrecy of their relationship is ruining her life.
So, for all of his big talk against the institution of marriage, it's Vronsky who's insistent that Anna divorce Karenin so that she can marry Vronsky. And after they leave Karenin to head for Italy to recover Anna's health, he goes so far as to resign his commission from the army entirely:
To decline a dangerous assignment to Tashkent would have been, to Vronsky's former way of thinking, disgraceful and impossible. But now he declined it without a moment's reflection and, noticing the disapproval of his act in high places, he at once resigned his commission. (4.23.34)
Where Vronsky would once have been eager to take this military station, he's now lost himself so much in Anna that he leaves the army without thinking twice. Anna has transformed him, in much the way that Kitty transforms Levin—but Vronsky's transformation isn't necessarily a good one. Where some transformations (such as that of Kitty and Levin) make those involved stronger, Anna and Vronsky keep feeding off one another to make things worse.
At the beginning of Anna Karenina, Vronsky was completely invested in Petersburg high society and in his own career. And while he loses everything in Anna, it turns out that their mad love isn't sustainable. Without the support of friends and family, in isolation, Anna starts to spiral downwards and takes Vronsky with her.
The final bone of contention between the lovers is precisely this issue of Vronsky and his role in society. Even after resigning his commission and leaving Petersburg for a while with Anna, Vronsky can still move in the same social circles he used to visit, while Anna's life has completely changed.
It isn't hard for Vronsky to continue life—or some aspects of his life—as he knew it before Anna. After all, Vronsky's a landowner and a well-liked guy around town. He's having some money troubles, but nothing that would get him barred from hanging out with the people he used to know. Vronsky even attends the provincial elections alongside Levin, Sviyazhsky, and Oblonsky in Part 6. What's more, Vronsky's able to restart his career: he knows the right people to suck up to in the new liberal party when he's attending the elections, and things are looking up for his ability to make something of himself again.
But Anna can't join in with her old friends. Even Princess Betsy, who's been having an affair with a guy named Tushkevich on the side for ages, won't visit Anna on the grounds that it would be social suicide. Things really come to a head in Part 5, Chapter 23, when Mrs. Kartasov at the French opera won't even sit in the box next to Anna's box. Anna can't even go to the opera without getting pointed at and humiliated by Petersburg's elite.
After this disaster, all Anna can see is that Vronsky's out, carousing with his friends, while she's stuck at home with a baby she doesn't love and nothing else to do. Vronsky may have given up his army commission and changed his life for her, but that's not enough. Anna can't see past the unfairness of the fact that Vronsky can have a life outside of their relationship, and she can't. It's her vulnerability as a woman in conservative Petersburg society that contributes, as much as anything, to her jealous breakdowns, accusations, and finally, to the suicide that leaves Vronsky a shadow of the man he once was.