One of the themes in The Great Gatsby is the quest for the American Dream. In this context, the American Dream generally means that an individual can achieve a level of success regardless of his/her background. This is why the "rags to riches" motif is so popular in American culture. As it is stated in the Declaration of Independence, all men (modernized as "people") are "endowed by their Creator with certain unalienable rights, that among these are Life, Liberty and the pursuit of Happiness. In this novel, the American Dream is the pursuit of happiness as personal, romantic, financial, or social success.
Buying into the possibility of the American Dream, Gatsby embarks on a rags to riches journey and with it comes the reinvention of his own persona. Saying he comes from wealthy parents and was educated at Oxford, Gatsby rewrites his past in order to secure dreams in his future. He achieves a high social position and is not held back by his humble beginnings because he has erased them and written himself a new history. Therefore, Gatsby didn't become successful despite his humble background. He became successful by erasing it and appropriating a background that would give him more opportunity to succeed. Although Gatsby does these things in a naive spirit of idealism, he essentially acknowledges that one can not achieve the American Dream from any background. He felt he had to change his background. This suggests that the American Dream, or achieving an idealistic human fulfillment, is not equally accessible to everyone.
Even when Gatsby does reunite with Daisy, she could not possibly live up to his idealization of her, thus leaving him with partial success, partial emptiness:
There must have been moments even that afternoon when Daisy tumbled short of his dreams—not through her own fault, but because of the colossal vitality of his illusion.
The American Dream and/or the pursuit of happiness are worthwhile. Fitzgerald criticizes the corruption of the American Dream. He criticizes the situation in which success is more accessible to people like Tom Buchanan. The situation is also corrupt when Gatsby, a man of humble beginnings, must lie and engage in bootlegging and gambling in order to increase his advantage in achieving success.
The novel suggests that the search for human fulfillment is a glorious goal, but when the (social/cultural) conditions for achieving the goal are tied up with corruption, the goal is going to be incomplete; thus, achievable but with some amount of emptiness. Gatsby had to erase his past (a void - emptiness). Tom is financially and socially successful but he is a racist, adulterer, and a manipulator: hardly a successful person in the ethical sense. Therefore, Tom is not completely successful; he is financially successful but as a person, he's flawed and therefore unsuccessful in that respect.
When the conditions favor those born into wealth, and those who succeed via illegal or unethical means, then the acquiring of success is contaminated and will likely end incomplete. The average individual is more likely to genuinely achieve the American Dream (fulfillment) when the conditions provide the opportunity to anyone, regardless of background.