Jay Gatsby's sole purpose in life is to reinvent himself in a way that will allow him to pursue and be accepted by Daisy. He has gone to incredible lengths and largely succeeded in the sense that he has earned a ton of money and set himself up in an incredible mansion and thrown lavish parties.
Yet something is missing and he is not entirely accepted by the people like Tom who come from real money. Something about the way that he earned his money, or perhaps even the fact that he earned it turns them off to him and he cannot truly realize his American dream, the dream of reinvention. Coupled with the way that his wealth and his life are also taken away from him by evil and bad luck, he can be seen as a representation of the failure of the American Dream.