The proverbial American Dream is treated as a complex and vague motivation for improvement in the novel, easily corrupted and distorted, easily made to serve fantasy instead of creating new realities.
The essence of the American Dream, as it exists as a broad conception of self-determination, relates to the notion of financial upward movement. Gatsby presents a clear picture of this idea.
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.
In his journal where he writes down his personal resolves as a young man, Gatsby communicates his yearning for self-improvement. He plans to save money and be better to his parents. His resolves are humble and practical.
Despite Gatsby's great success, we have to see him as an impostor in the "high class" world he entertains at his parties. He is aligned with the achievement of the American Dream but also serves as an example of how it can be distorted.
When we meet him Gatsby is no longer humble or practical. Achieving great wealth, Gatsby cannot is not the person he once was, nor is he the person he pretends to be. He is lost somewhere between James Gatz and Jay Gatsby.
The narrative presents this state of affairs with moral implications. Tied to underworld figures (the people who fix the World Series, etc.), Gatsby has lost something which may not be commensurate with what he has gained.
The self-improvement aspect of Gatsby's American Dream is not at all guaranteed, it seems, as a result of financial improvement. Instead of altering his reality, Gatsby slips into a belief in fantasy, insisting on the impossible and, naturally, falling short of his vision.