I was lying when I said just now that I was a spiteful official. I was lying from spite. (1.1.5)
This is some classic existential absurdity going on here. If he's lying from spite, it must mean that he's spiteful. The Underground man introduces this type of logic (or illogic?) early in the text, setting the stage for the "reasoning" that is to follow.
How much better it is to understand it all, to recognise it all, all the impossibilities and the stone wall; not to be reconciled to one of those impossibilities and stone walls if it disgusts you to be reconciled to it; by the way of the most inevitable, logical combinations to reach the most revolting conclusions on the everlasting theme, that even for the stone wall you are yourself somehow to blame, though again it is as clear as day you are not to blame in the least (1.3.7)
Jean-Paul Sartre would later continue this idea of radical personal freedom; every man is free – completely. It's not that you're not prevented in any way from committing murder; technically, you can. It's just that you don't want to face the consequences. The downside, however, is that you are responsible for everything that happens to you. When the Underground Man talks about being to blame for everything – even the laws of nature that you really can't do anything about – he is creating an the early premise for this later concept.
[…] to sink into luxurious inertia, brooding on the fact that there is no one even for you to feel vindictive against, that you have not, and perhaps never will have, an object for your spite, that it is a sleight of hand, a bit of juggling, a card-sharper's trick, that it is simply a mess, no knowing what and no knowing who, but in spite of all these uncertainties and jugglings, still there is an ache in you, and the more you do not know, the worse the ache. (1.3.7)
Some existentialists believed that there was no such thing as objective truth. Similarly, the Underground Man suffers from the uncertainty of his knowledge of everything. He constantly second-guesses himself.
I repeat, I repeat with emphasis: all "direct" persons and men of action are active just because they are stupid and limited. How explain that? I will tell you: in consequence of their limitation they take immediate and secondary causes for primary ones, and in that way persuade themselves more quickly and easily than other people do that they have found an infallible foundation for their activity, and their minds are at ease and you know that is the chief thing. To begin to act, you know, you must first have your mind completely at ease and no trace of doubt left in it. Why, how am I, for example to set my mind at rest? Where are the primary causes on which I am to build? Where are my foundations? Where am I to get them from? I exercise myself in reflection, and consequently with me every primary cause at once draws after itself another still more primary, and so on to infinity. […] So you give it up with a wave of the hand because you have not found a fundamental cause. (1.5.1)
According to many existentialists, the Underground Man's reasoning is right: since there is no objective truth, and since the universe is without reason, you can never justify any of your actions. However, an existentialist would condemn his conclusion of inaction. Just because you can't justify your actions doesn't mean you shouldn't act anyway.
He loves the process of attaining, but does not quite like to have attained, and that, of course, is very absurd. In fact, man is a comical creature; there seems to be a kind of jest in it all. (1.9.2)
Again we're back to the notion of absurdity; there's something very dark about the lack of reason in the world, but also something comical.
Another circumstance, too, worried me in those days: that there was no one like me and I was unlike anyone else. "I am alone and they are everyone," I thought – and pondered. (2.1.3)
Jean-Paul Sartre would later introduce the idea of the subject-object problem. Every man sees himself as subject and others as objects. He can never really conceive of another man as being a subject himself. The Underground Man is getting at something similar here; he sees a clear division between himself and everyone else. Those are two distinct categories in his mind, and he makes no attempt to see another human being as, well, a real human being.
Anton Antonitch Syetotchkin […] was the only permanent acquaintance I have had in my life, and I wonder at the fact myself now. But I only went to see him when that phase came over me, and when my dreams had reached such a point of bliss that it became essential at once to embrace my fellows and all mankind; and for that purpose I needed, at least, one human being, actually existing. I had to call on Anton Antonitch, however, on Tuesday – his at-home day; so I had always to time my passionate desire to embrace humanity so that it might fall on a Tuesday. (2.2.3)
This is quite ridiculous. The Underground Man has to time his desire for contact, to schedule his bouts of human feeling to fall on any given Tuesday. This is another case of radical absurdity.
I'll go this minute!"
Of course I remained. (2.4.44-5)
If you've read Waiting for Godot (you can find it on Shmoop), then this line will look very familiar to you. Beckett's characters would suffer from the same paralysis of inaction as the Underground Man. In fact, they repeatedly say, "Let's go" and proceed not to move, just as is the case here. Now can you start to see why Notes is called an "overture" for existentialism?
There is no explaining anything by reasoning and so it is useless to reason. (2.9.33)
Existentialists for the most part agree in the first half of this statement; it is the second half where they beg to differ.
[The notes of this paradoxalist do not end here, however. He could not refrain from going on with them, but it seems to us that we may stop here.] (2.10.24)
This endnote makes an important point: the Underground Man declares he will stop writing, but then proceeds not to. This means not only can he not start an action, he can't seem to stop one either. Sound familiar? That's because he said as much in Part I, Chapter V: "All my life I have been able neither to begin nor to finish anything." Or, to put it another way, he's suffering from inertia: the tendency for things to stay as they are.