Nearly all the main characters in Fitzgerald's The Great Gatsby are trying to better their lives, which is, in essence, trying to achieve the American dream. Whether they achieve that goal or not varies by degree, but in the broadest terms, none of them have much success.
Jay Gatsby - achieves the American Dream in terms of money, which is what the young Jay Gatz initially wants to do. Once he falls in love with Daisy, though, his dream changes and he wants only to be in a position to have her love him. In that he fails, as she doesn't love him enough to abandon her life.
Tom and Daisy Buchanan - live the American Dream in that they are careless people who get what they want and discard the rest. Their failure is a moral failure.
Jordan Baker - is living the American dream of success in her chosen field. Her success is tarnished, however, because she cheats.
Myrtle Wilson - has a more modest American Dream, as she simply wants to escape the life she chose and thinks money will make her happy. She did escape (until the end) but she was even more unhappy when she had to go back to her home.
George Wilson - has a very modest American Dream, as he just wants more than he has. His life is an abject failure, as he achieves neither financial success nor personal happiness.
In short, virtually none of the main characters in this novel achieve their version of the American Dream.
There are many quotes reflecting this theme in The Great Gatsby. I will provide a few. Once you see some examples, perhaps you can find even more on your own.
In Chapter I, Nick is describing Tom and says,
...[B]ut I felt that Tom would drift on forever seeking wistfully for the dramatic turbulence of some irrecoverable football game (10).
This quote shows that Tom will continue to seek the glory of his lost youth, showing his failure to achieve the American Dream.
In another scene, in Chapter II, Myrtle, Tom's girlfriend, talks about her marriage to Wilson. She says,
I married him because I thought he was a gentleman.... I thought he knew something about breeding, but he wasn't fit to lick my shoe (39).
Later on in the same chapter, Myrtle's sister says,
She really ought to get away from him.....They've been living over that garage for eleven years (39).
This quote demonstrates the attempt and failure to achieve the American Dream for two characters, Myrtle and her husband.
There are few if any characters in the novel who have not tried and failed. See if you can find more.