The American Dream is rather difficult to define. I think that some premises can be agreed as a part of it. The idea of "having it all" is a significant part of the American Dream in early the mid 20th Century America. This is seen in the demand for economic prosperity, lavish spending on homes and objects, and the psychological link that more money can translate into more happiness. Part of the American Dream that Fitzgerald criticizes is the direct variation between both ends. It is in this domain where Fitzgerald's work is the most powerful in that it makes very clear that money and happiness are two separate entities and that the presence of the former does not automatically translate into the latter. In this light, the work reflects how many people of the 1920s, the Jazz Age, believed that wealth acquisition and display was vitally important in helping to define their success.
Fitzgerald's thesis in The Great Gatsby is that the American dream is fleeting and a fantasy for most. That Ameria looks great from a distance, but if you get close enough you will see its obvious flaws. America is really two Americas: one idealistic (Gatsby's America) and one corrupt (the Buchanans' America).
In the end, we are like Nick, caught in the middle, attending a funeral, left only with a memory of Gatsby. By killing his idealistic young hero, Fitzgerald seems to be saying the idealistic young days of the American dream are over, and we are left with the corrupt America where the rich abuse their privilidge and victimize the poor. Or, maybe it's always been corrupt, and we have only been deceiving ourselves in thinking it's been otherwise.
Examples are mainly found in chapter 9 where Nick says:
And as I sat there, brooding on the old, unknown world, I thought of Gatsby's wonder when he first picked out Daisy's light at the end of his dock. He had come such a long way to this blue lawn, and his dream must have seemed so close he could hardly fail to grasp it. But what he did not know was that it was already behind him, somewhere in the vast obscurity beyond the city, where the dark fields of the republic rolled on under the night.
Also important is the critical essay on Enotes, which says:
The ideal of the American Dream is based on the fantasy that an individual can achieve success regardless of family history, race, or religion simply by working hard enough. Frequently, “success” is equated with the fortune that the independent, self-reliant individual can win. In The Great Gatsby, Fitzgerald examines and critiques Jay Gatsby’s particular vision of the 1920s American Dream. Though Fitzgerald himself is associated with the excesses of the “Roaring Twenties,” he is also an astute social critic whose novel does more to detail society’s failure to fulfill its potential than it does to glamorize the “Jazz Age.”
The American Dream is embodied in home ownership of late. In the 19th century, having a lot to homestead was the American Dream. In Gatsby, Gatsby's mansion certainly demonstrates success. His father's tattered picture of Gatsby's home, as well as the gift of a home from Gatsby to his father all align with this American Dream.
Materialism, fame and success in love are all pieces of the American Dream. Each of these is demonstrated in the relationship between Gatsby and Daisy. The problem though with any dream is that it can be the alter ego of reality. This dream when approaching reality destroyed it. I think this is a value that Gatsby is trying to teach.