Satire, Fantasy, Parable
From start to finish, there's no escaping the snark in this story. To begin with, there's Poprishchin scorning everyone and everything with his ridiculous exaggerations. Remember what he has to say about Sophie's fiancé, Teplov? "So what if he's a kammerjunker […] He's not going to have a third eye on his forehead because he is a kammerjunker" (9.1). Or what about the director? "He's a doornail, not a director. An ordinary doornail, a simple doornail, nothing more" (13.1). Gogol manages to satirize everyone from elite Russian ladies (see "The Dogs" under "Symbols, Imagery, Allegory") to the upper echelons of the Russian nobility (2.1) the Jews (1.1), the provincial government (1.1), the French (2.1), merchants (4.1), journalists (4.1), women (7.1), the Finns (7.1), Muslims (13.1), and the English (18.1).
But the real joke is on Poprishchin: when we're not laughing with him, we're laughing at him (and the lower ranking Russian civil servants he represents), doing things like copying verses by a mediocre poet (2.1) and thinking he's so cultured.
This story is also fantastic. Okay, we admit that dogs might speak (to our hearts), but no way they're writing letters to one another.
And finally, we can think of the whole thing as a parable for your average Russian Yuri, going through life being confused, hopeless, and miserable.