First of all, we're going to talk about the tone of Notes from the Underground – not the tone of the Underground Man himself. The difference is that we're looking at the work as a whole and addressing Dostoevsky's attitude, not just the bitter and angry attitude of his main character. Now, the question of whether or not the Underground Man is Dostoevsky himself is a big can of worms. To address this question means to go into Dostoevsky's other works, mess around with his biography, read his letters and personal accounts – we're talking about a major, major project.
Still, if you don't want to do this, you can look at Notes as it stands alone. If the question really interests you, check out the beginning and end notes from the author. What hints does Dostoevsky give about his view of the Underground Man? Does he try to distance himself from his character? We'll let you take care of that while we address what appears to be the author's tone as distinct from the tone of his character.
We'll start with the easy part, our "darkly comic" label above. You may have laughed a few times reading Notes from the Underground. If you're reading the Constance Garnett translation (which we read), the humor might not come across as clearly as it does with, say, translator Michael R. Katz. Check out this passage from Katz's translation: "Well, even if you reach the same result […], at least you'll be able to flog yourself from time to time, and that will liven things up a bit. Although it may be reactionary, it's still better than nothing."
Now let's get to this "polemic" word. A polemic is a written assault. When someone writes something opinionated and theoretical that you think is absolutely wrong, you can respond to it by writing something equally opinionated and theoretical that basically says, "Not a chance, you got it all wrong." And that's called polemic. (If you want the specifics about which theories Dostoevsky was refuting, check out Shmoop's "Overview." For now we're just going to talk about his scathing approach.)
How can we tell it's polemic? Look at the Underground Man's fantasy of being a professional "aesthete" (a person focused on the artistic qualities of everything). "An artist, for instance, paints a picture worthy of Gay. At once I drink to the health of the artist who painted the picture worthy of Gay, because I love all that is 'sublime and beautiful.' An author has written As you will: at once I drink to the health of 'anyone you will' because I love all that is 'sublime and beautiful.'" This is Jon-Stewart-quality material. Dostoevsky is clearly mocking the fat (his description, not ours), lazy people who "appreciate art" for a living. Even the phrase "sublime and beautiful" comes from Immanuel Kant, one of the men against whom Dostoevsky rails. Dostoevsky's tone, in its exaggeration and incisive bite, screams "polemic."
Of course, there are those who think it's not polemic, that Dostoevsky actually wasn't writing in response to Chernyshevsky (the Crystal Palace guy) and all the other thinkers/writers/philosophers to whom he refers.For more perspectives on this issue, please see websites such as this and this.