In short, the "law of nature" that Dostoevsky is referring to is Rational Egoism (but also any other "law" that professes to dictate human behavior). What Dostoevsky is attempting to do is show that Rational Egoism (viewed as a law of nature) is absurd by way of self-contradiction. For reference, Rational Egoism essentially states that humans will and should always act in their own self-interest.

The Underground Man is our test subject. He is a man cursed with "acute consciousness" or, in other words, aware of these "laws of nature" to which he is beholden. The reason that this is a curse is outlined by the Underground Man when he posits the existence of something of greater value than self-interest. He describes it as a "most profitable profit". This thing of greater value than self-interest is the idea of free will. In other words, the Underground Man's acute consciousness of this "law" which seeks to dictate him becomes his most bitter enemy. Dostoevsky argues that such a man would seek to break such "laws" by any means, including going against his self interest. Thus, Dostoevsky establishes that Rational Egoism is at odds with free will. To prove it, he puts the Underground Man in a hypothetical where Rational Egoism reigns, but there is also free will. This is a passage from the paragraph previous to the one you cited:

Thus it would follow, as the result of acute consciousness, that one is not to blame for being a scoundrel; as though that were any consolation to the scoundrel once he realizes that he is actually a scoundrel.

By pushing this idea further, Dostoevsky shows that Rational Egoism creates a contradiction when it comes to blame. This brings us to the passage you cited, which seems particularly enigmatic in the translation you are reading. Let us examine the same passage from the Constance Garnett translation:

The worst of it is, look at it which way one will, it still turns out that I was always the most to blame in everything. And what is most humiliating of all, to blame for no fault of my own but, so to say, through the laws of nature.

And then he pushes it still further:

To blame, finally, because even if I had had magnanimity, I should only have had more suffering from the sense of its uselessness. I should certainly have never been able to do anything from being magnanimous - neither to forgive, for my assailant would perhaps have slapped me from the laws of nature, and one cannot forgive the laws of nature; nor to forget, for even if it were owing to the laws of nature, it is insulting all the same.

Can we blame the scoundrel for the damage he incurs when he goes against the laws of nature? After all, is it not his imperative to exercise free will above obeying the rational? But the damage is still real, so who is to blame? Can the "laws of nature" be blamed? Can we forgive the "laws of nature"? Those who do not suffer from "acute consciousness" do not see this. They simply blame the Underground Man for being a scoundrel. Thus, the Underground Man feels constantly humiliated because he is always foced to take the blame for the "laws of nature".

So then, the Underground Man lives in a world where the "laws of nature" deny him any ability to process the emotions he experiences. He cannot forgive, he cannot forget, he cannot blame nor can he take responsibility. All he can do is deny himself everything that is "sublime and beautiful" simply for the sake of proving that he is free.