You could take this from two perspectives: Nick, the tool narrating character reporting situations that reflect the author's positions, or Nick Carraway, the character.
Fitzgerald positions Nick throughout the story to report and judge characters who all were in pursuit of the American Dream. Through his experiences with Daisy, Tom, and Gatsby, I think it's clear that money is not it. This may be Fitzgerald's perspective. His experiences with Miss Baker and folks who go to parties and attach themselves to Gatsby superficially likewise demonstrate an unfulfilled American Dream.
Nick, the character watches Gatsby's life and by the end must come to the conclusions that happiness comes from living and working in give and take relationships. Gatsby had no one concerned enough to come to his funeral and Nick experienced the shallow life Gatsby must have had even though he has so much. This had to speak volumes to Nick. He might have also noticed that Gatsby didn't contribute to society at large. If that had been true, a much greater ripple would have been made because of his death. These two truths might be what Nick sees as elements of the American Dream.
In The Great Gatsby, Nick (Fitgerald) says the American Dream is both a promise and a false promise. Fitzgerald's novel both confirms and denies the American Dream.
Gatsby embodies both the promise and the false promise. He lives the American Dream up to a point, going from rags to riches. They say there are no second acts in America, but Gatsby reinvents himself. However, he does so by illegal means.
Even though the American Dream is very Machiavellian (the ends justify the means), it is exclusive for only those who feel they deserve it (like Tom). Gatsby achieves it, and Nick says it didn't matter how her achieved it (bootlegging), but Tom easily snatches it away from Gatsby after Gatsby tries to steal his wife. In the end, the American Dream is fleeting for those who come from have-not backgrounds.
Gatsby's mistake is that he tries too hard to legitimize himself and his dream: he denies his past too much. He tries to appear to be like the old rich, like Tom. He even tries to take Tom's wife. In the end, Tom makes sure that Gatsby pays dearly. Tom uses a man of no dreams (George) to murder a man of only dreams (Gatsby).
Fitzgerald presents the myths of America and then explodes them. He demonstrates how things look promising from a distance, but dont look too closely--like Nick, who describes himself as both within and without. So says Enotes:
Gatsby represents the American dream of self-made wealth and happiness, the spirit of youth and resourcefulness, and the ability to make something of one's self despite one's origins. He achieved more than his parents had and felt he was pursuing a perfect dream, Daisy, who for him embodied the elements of success. Gatsby's mentor, Dan Cody, was the ultimate self-made man who influenced Gatsby in his tender, impressionable youth. When Gatsby found he could not win Daisy's love, he pursued the American Dream in the guise of Cody. Inherent in this dream, however, was the possibillty of giving in to temptation and to corrupt get-rich-quick schemes like bootlegging and gambling. Fitzgerald's book mirrors the headiness, ambition, despair, and disillusionment of America in the 1920s: its ideals lost behind the trappings of class and material success.