Yes, Gatsby exemplifies the American Dream. The American Dream can be looked at in two different ways.
1) The sense of limitless possibility at teh heart of America in that every American can make something of themselves so long they make use of their natural talents and hard work and the likes. It is the embodiment of human potential and this is seen in Gatsby's "extraordinary gift for hope". Gatsby has hope that he can recapture/relive the past with Daisy Buchanan and he believes in this illusion so much that he has made Daisy the object of his dream. And he lives his life in such a way that Daisy is the basis. There is the green light at the end of Daisy's dock, the lavish parties hoping that Daisy may someday return, etc. Everything he does is for the belief that Daisy may declare true love for him and they may live happily ever after.
2) The materialistic aspect of the American Dream where everyone's main focus is on getting rich - which ever way. Gatsby had a dream (you can even argue, a hope) to be a better man. To make something of himself. Rise above his father's material status because he was/is a nobody and also he wanted to get rich for this "excitingly desirable" girl, Daisy. He does this through corrupt means and this is the negativity portrayed in the American Dream in that people are so engrossed in getting materialistc wealth that they end up failing in terms of morality.
There is an idea of moral failure and material success (and vice versa) and Gatsby has failed morally but succeeded materially (contrasting with his father who is poor but presumably generally good.)