Since Notes from the Underground is told in the first person, most of what we know about the Underground Man is the result of hearing his thoughts and opinions. And oh, does he have a lot of them. His thoughts on romanticism reveal his own romantic idealism; his stories reveal his jealousy and spite; his discussion of his attacks of the "sublime and beautiful" reveals his fickle extremism.
As you might have noticed, the Underground Man has no name. He's not even called "the Underground Man" in the text; critics just started referring to him that way, maybe because "that guy who tells the story from under the ground whose name we don't know" was getting cumbersome. By not having a name, he seems to us almost less than human. His identity? Unclear.
Social Status is, for the Underground Man, an important indication of character. Liza is a prostitute, therefore she is ignorant, degraded, and in need of saving. Zverkov is a feudal lord, so he's condescending, vain, and superior. The Underground Man judges himself by the same standards; he's impoverished, therefore he's shameful and disgusting.
The Underground Man is basically defined by his inability to act. That's why he's living Underground, that's why he can't stop writing, that's why he's a disembodied voice in Part I instead of a full-blown person, with a real identity and everything. Liza, too, is defined by her actions: when she comforts the Underground Man despite his cruelty, we know more about her than we could have from earlier scenes.