One book that is often discussed in the context of the American Dream of self-determination and upward mobility is The Great Gatsby by F. Scott Fitzgerald. This 1920s novel deals with a number of issues that relate to the American Dream with a character in Jay Gatsby who remakes himself from a poor Midwesterner into a very wealthy man throwing parties for the rich elite. Gatsby's ambitions exist next to the ennui and moral lassitude of the "old money" crowd -- people like Tom and Daisy Buchannon.
Yet, despite the moral corruption that characterizes the wealthy elite, figures like Nick and Gatsby continue (for a time) to pursue a dream of social mobility and hanker for a true achievement vis a vis class and social connection. They stop their striving only when it proves fatal to continue (literally and figuratively).
"Fitzgerald was also a thoroughly romantic artist in the most traditional sense; and, for him, women like Daisy represented the deepest seductive power of the Dream as well as its greatest dangers" (eNotes).
In these ways and others, The Great Gatsby is a novel of the American Dream.
Death of a Salesman by Arthur Miller and Babbitt by Sinclair Lewis are also works of narrative fiction that relate to the American Dream (one a play and the other a novel, respectively).
Miller's play deals with achievement and ambition and the delusions that can accompany them. With a simplistic but almost ritualistic definition of success, Willy Loman imbues his two sons with the notion that one must be "well liked" to get ahead in the world of business and this narrow presumption keeps Willy's ambition painfully fresh and keeps him from realizing all the things he actually does accomplish in his life.
Babbitt deals with a specifically corporate/white collar American Dream, placing the generic value system that advises one to always advance into conflict with a wider set of value systems prevalent in America.
While the works mentioned above deal with a conventional view of the American Dream (linked to ideas of money and social success), there are other ways to define the American Dream and other works of literature that articulate those views.
Walt Whitman's Leaves of Grass is a collection of poems that passionately rhapsodizes on ideas of individualism, democratic spirit and dynamic intellectual life. For Whitman, these are qualities that define America and its promise.
Thoreau's work, known sometimes as Walden, explores ideas that are sometimes similar to Whitman's and to those of Ralph Waldo Emerson.
"One reason that Walden is exemplary as a work of Transcendentalism is that the book makes the idealistic assumption that there is a true self to discover" (eNotes).
What do Transcendentalism and the American Dream have in common? They both see the individual self as a perfect-able, discover-able thing. This view of the self is one of the foundations of the American Dream as it is inflected in The Great Gatsby and Death of a Salesman.
Other works that deal with American ambition, social climbing, and/or a Transcendental view of the pursuit of self:
On the Road by Jack Kerouac
All the King's Men by Robert Penn Warren
An American Tragedy by Theodore Dreiser