Really, in the novel, the American Dream is defined as a fantasy, an impossibility.
It isn't just about getting money. If it were, then Gatsby would be living the dream, and he most certainly is not. The American Dream refers to the idea that, in this country, if one simply works hard enough, one can prosper and become successful. Gatsby tries to work hard, but doesn't begin to acquire the financial position necessary to associate with Daisy honestly. So, he turns to illegal activities in order to reach her, but once she learns that he's made his money as a bootlegger, a criminal, she no longer wants to be with him. The American Dream requires that one do honest, legal work to achieve success. The irony is that Gatsby couldn't get there that way either.
Likewise, George Wilson attempts to do honest, legal work to achieve the American Dream, but he's no closer to achieving it than a young Gatsby had been. He's still jerked around by Tom Buchanan, cheated on by his awful wife, and made sick by the news of her infidelity. He eventually realizes the futility of attempting to reach the dream, and he kills himself as well as the person he blames for the loss of his wife.
Gatsby can't prosper unless he does so illegally, and Wilson can't prosper doing legal work -- the idea that one need only work hard enough is revealed to be a lie. All of the people who come from nothing in the novel and attempt to achieve success -- Gatsby, Wilson, and Myrtle -- die. It is, thus, an impossibility.