Like many American writers, F. Scott Fitzgerald created characters and themes which explored the American Dream. In The Great Gatsby, we can see numerous themes relating to the American Dream of self-reliance, self-creation, and the attainment of wealth.
Examples of the American Dream gone awry are plentiful in The Great Gatsby: Meyer Wolfsheim's enterprising ways to make money are criminal; Jordan Baker's attempts at sporting fame lead her to cheating; and the Buchanans' thirst for the good life victimizes others to the point of murder.
Jay Gatsby is, of course, the most prominent example of a character in pursuit of the American Dream, as he makes his way from a relatively poor Midwestern young man to a very wealthy, easterner with a closet full of beautiful shirts and a vast mansion.
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 Gatsby's relationship to the American Dream we can see that on the one hand this dream is achievable. If it was only wealth that Gatsby wanted, he would have achieved his dream. Yet, the dream is more complex than one for simple wealth. Both Gatsby's dream and the American Dream are also about becoming a new kind of person and leaving the past behind.
This aspect of the dream is not attainable for Gatsby, no matter how hard he tries. His past as a Gatz, as a romantic "loser", and as a loner is still with him to the end of the story in a very literal way.
We can say then that the American Dream of self-creation is counter-balanced a theme in Gatsby of self-deception. There is a question integrity implicit in the figure of Gatsby, a man who seeks to completely divorce himself from the past, to veil that past in a lie.
This question of integrity can, perhaps, be extended to the American Dream at large.
To quickly suggest some other writers who also have examined and questioned the American Dream, you may want to look at Herman Melville, Mark Twain, Kurt Vonnegut, and Don DeLillo.