The concept of the American Dream is...
...based on the fantasy that an individual can achieve success regardless of family history, race, or religion simply by working hard enough.
The reason this ideal became so popular is because at one time in the United States—viewed as "the land of opportunity"—people could come from all over the world, open a business and become financially successful. (The idea of "opportunity" in the U.S. today is radically different: today it appears to be more closely equated with religious and political freedom; financial gain is also possible for those living in countries with few jobs and a weak economy.)
However, in F. Scott Fitzgerald's The Great Gatsby, in the time period in which it is set, the potential for financial success still existed, and Gatsby cashes in on this. To understand how he achieves the American Dream, one only need look at his background growing up and compare it to his financial position and lifestyle when Daisy comes into his life for the second time.
His parents were shiftless and unsuccessful farm people—his imagination had never really accepted them as his parents at all. [He saw himself as] a son of God—a phrase which, if it means anything, means just that—and he must be about His Father's business...So he invented just the sort of Jay Gatsby that a seventeen-year-old boy would be likely to invent, and to this conception he was faithful to the end.
This quote introduces the life Jay Gatsby was born into: his parents were not just farmers, but "shiftless" people, meaning they were lazy. On a farm, this would spell disaster, and living off of a poorly tended farm would be a struggle for survival at best.
However, Gatsby is savvy enough to know that he needed to change his image before he would be able to seize the opportunity to create his dream. That is where "James Gatz" became Jay Gatzby. Nick Caraway makes note of Gatsby's success. Gatsby is his neighbor, and Nick observes the opulent lifestyle Gatsby has. There is always music playing, champagne is served, and people spend long casual hours (as only the rich can afford) to stroll through gardens and lie on his beach. Gatsby has a motorboat, and aquaplanes come and go.
On week-ends his Rolls-Royce became an omnibus, bearing parties to and from the city between nine in the morning and long past midnight, while his station wagon scampered like a brisk yellow bug to meet all trains. And on Mondays eight servants, including an extra gardener, toiled all day with mops and scrubbing brushes and hammers and garden-shears, repairing the ravages of the night before.
It would seem that Gatsby has achieved his dream of financial success. He has been so successful in reinventing himself that he receives compliments wherever he goes. Nick meets Mr. Wolfsheim while spending time with Gatsby. Woflsheim completely believes the facade that Gatsby has created:
I made the pleasure of his acquaintance just after the war. But I knew I had discovered a man of fine breeding after I talked with him an hour. I said to myself: 'There's the kind of man you'd like to take home and introduce to your mother and sister.'
This kind of behavior would express someone's full approval, for only the deserving were taken home to meet the women. Gatsby seems to have it all. Without Daisy, however, it is just an illusion of success for him.