While I cannot give you a description of Nick Carraway for every chapter, as you asked, I can give you an overview of his character throughout The Great Gatsby, by F. Scott Fitzgerald.
Nick is from the Midwest, and he is not moved by all the money he sees displayed by the Buchanans and Gatsby--a bit dazed, perhaps, but not moved. Though he is in New York to work at the Stock Exchange and, presumably, make some money, he is not greedy and in fact turns down several "business opportunities" throughout the novel.
Nick is the narrator of the novel, so it is through his eyes and voice that we see and hear the events of the novel unfold. Though in the opening paragraphs Nick claim he is "inclined to reserve all judgments," he does make many of them throughout his time in New York. This causes readers to question Nick's reliability as a narrator.
He graduated from New Haven in 1915 and then went off to fight in the "Great War" before eventually landing in the little bungalow next to Gatsby's glittering mansion in West Egg, New York, in the spring of 1922. His father has agreed to pay for Nick to come learn the "bond business" for a year.
Anything we learn about Gatsby we learn through Nick, and he foreshadows that learning early in the novel.
Gatsby, who represented everything for which I have an unaffected scorn. If personality is an unbroken series of successful gestures, then there was something gorgeous about him, some heightened sensitivity to the promises of life, as if he were related to one of those intricate machines that register earthquakes ten thousand miles away. This responsiveness had nothing to do with that flabby impressionability which is dignified under the name of the "creative temperament"-it was an extraordinary gift for hope, a romantic readiness such as I have never found in any other person and which it is not likely I shall ever find again. No--Gatsby turned out all right at the end.
Here we learn that Gatsby was everything Nick hated, but we will only learn the specifics as the story unfolds. Gatsby is a liar (Nick hates liars), and Gatsby is an unscrupulous businessman with "connections," something else Nick disdains. Gatsby spends time and money just to impress people, something Nick is certainly not interested in; however, Nick gets a new understanding of Gatsby's parties once he learns about Gatsby's desperate love for Daisy. Nick watches Gatsby, and what he sees in him, his heart and his "romantic readiness," move Nick to change his opinion of Gatsby. The last thing Nick says to Gatsby is: "They're a rotten crowd..... You're worth the whole damn bunch put together."
When Gatsby dies, Nick finally realizes he is the only one who saw anything of value in Gatsby. "I found myself on Gatsby's side, and alone." Nick is eventually disgusted with all of these dishonest people--Daisy, Tom, Jordan--that he leaves New York.