Liza is a prostitute. If you're only going to know one thing about Liza, this should be it – especially because, in the Underground Man's eyes, this is her defining quality.
Other than that, she can seem a bit hard to pin down. She attracts the Underground Man without being particularly pretty. She is young, but with the "grave" nature of someone older. At first, Liza callously rejects the Underground Man's attempts to make it clear that, really, her life is awful and, in fact, is only going to get worse. "What's it to you?" she wants to know.
Of course it's not long before she's weeping into her pillow, so the Underground Man clearly got to her. (A horrific vision of your future would probably do the same to you.) But it's not until Liza comes to the Underground Man's home that we start to get some real insight into her character. She's nice. Wicked nice, in fact. We'll even go so far as to call her 'tremendously compassionate.' The Underground Man yells at her, tells her she's worth nothing, berates her life and choices, predicts her miserable death, and claims to have been mocking since they met. And yet, when he breaks down crying, Liza comforts him.
Now the interesting question when thinking about Liza actually has more to do with the Underground Man than it as to do with her. Why does he treat her this way?
Glad you asked! We (and the world of scholarly criticism) have a few theories.
We're not surprised by this one. The Underground Man claims that he can't find primary reasons for any action. If we ask why he treats Liza so badly, we can't really expect him to whip out a stellar justification. The answer would probably be inertia; he's going along with the status quo flow, and the river has been running in the direction of hurtfulness for as long as he can remember.
This would be even less surprising, considering how he says that "the world may go to pot […] so long as [he] always get[s his] tea." The Underground Man has made it quite clear that he loves pain and suffering, and that inflicting it on others (think of the toothache guy) is enjoyable. So if he's making Liza miserable, it's only because…he likes to make people miserable.
Again, this makes sense based on what we've seen. The idea of a wise older cultured man saving a prostitute is popular storyline in Russian Lit. (Just think Pretty Woman and you'll get the picture.) If you need proof, just look at the epigraph – Nekrasov's poem is about this sort of relationship. So the Underground Man is acting out one of his literary fantasies – just like he tried to do with the officer on the Nevsky (that fantasy was about revenge, not prostitutes, but it's another example of a romantic literary scene). If so, then what appears to be unabashed cruelty is really just tough love. The Underground Man posits this theory himself after Liza leaves: "Resentment – why, it is purification; it is a most stinging and painful consciousness! Tomorrow I should have defiled her soul and have exhausted her heart, while now the feeling of insult will never die in her heart, and however loathsome the filth awaiting her – the feeling of insult will elevate and purify her ... by hatred."
Yes, that's right – it's possible that the Underground Man really is in love with Liza. Who knows – maybe love is about suffering. Maybe to love means to tyrannize, to subjugate, to…wait a second. This is all sounding very familiar. See the following:
"With me loving meant tyrannising and showing my moral superiority. […] Love really consists in the right – freely given by the beloved object – to tyrannise over her. […] I did not imagine love except as a struggle. I began it always with hatred and ended it with moral subjugation, and afterwards I never knew what to do with the subjugated object."
Cut to the picture of Liza huddled up crying on the floor after their second time in bed. The Underground Man is flipping out because he feels "oppressed" by her presence and doesn't really know what to do with her. (Her = the subjugated object…see where we're going?)