What is the relationship between the first and second parts of Notes from Underground?

The first part of Notes from Underground takes the form of a character sketch written in first-person narrative. The unnamed narrator tells the reader in considerable detail what kind of a man he is. Fairly soon, we're under no illusions that this is a deeply conflicted, complicated man with numerous negative character traits as fascinating as they are repellent. The underground man represents a case study in nihilism, the idea that there is nothing of ultimate value in the world. Nihilism was prominent among disillusioned members of the Russian middle classes in Dostoevsky's day, and in his withering contempt for the norms and values of society, the underground man symbolizes many of his fellow countrymen.
In part two, the underground man fills in some details concerning his background. The style in which the second part is written is markedly different than that of the first. There, Dostoevsky employed an interior monologue, giving us privileged access to the narrator's numerous thoughts and feelings as they flowed rapidly through his consciousness. In the second part, however, he opts for a more conventional narrative style as the narrator recounts episodes from his earlier life, which chronologically pre-date part one of the book.
To a large extent, the first part provides the raw materials for the second. We already have some idea of what kind of a person the underground man is; the second part fleshes out the portrait of his personality, providing some clues as to how he got that way. One could also say that the first part deals with rather abstract ideas; this part of the book is the more overtly philosophical of the two. It is here that the underground man shocks us with his nihilism and total alienation from a society he clearly despises. The second part deals with reality at a more concrete level, with how the underground man lives out his often paradoxical ideas in his dealings with other people.
In presenting us with the contents of the narrator's fraught consciousness first, Dostoevsky skillfully whets the reader's appetite for more information. We may be repelled by this strange character, but there's no denying there's something intriguing about him. So we read on in anticipation, eagerly waiting to see how the underground man came to be in such a troubled state of mind.