If we define The American Dream in its most simplistic terms, it is the possibility that anyone from any background can become successful if they work hard and persist in the achievement of their goals.
From that perspective, Gatsby is emblematic of The American Dream. He came from a poor, rural background, worked hard, and later achieves a new identity as a rich, cosmopolitan man.
Because he has transformed himself, he believes that he is now worthy of Daisy's love. However, by making it so that Daisy still rejects Gatsby in favor of Tom, Fitzgerald exposes the hypocrisy of The American Dream. Contrary to what we tell ourselves, there is a class system in America that is not unlike the one which exists in England: the class you are born into is the one with which you will be associated.
Fitzgerald once said, "There are no second acts in American life." This idea is illustrated in his treatment of Gatsby and Daisy's relationship. She met him as a poor boy. His newly accrued wealth has not erased that image. He cannot redo his past with her. He may not have a second chance.