Is it possible to establish a literary-historical line about the American dream in American literature? How can we do that?

There are indeed several ways to do this. The first step in this task is conceptual clarification. One can either research the history of the term "American Dream" or one can try to define the American Dream as a concept, and then chart a history of its conceptual development independent of the specific uses of the phrase. 

The origin of the phrase is credited to James Truslow Adams' 1931 book Epic of America. One way...

There are indeed several ways to do this. The first step in this task is conceptual clarification. One can either research the history of the term "American Dream" or one can try to define the American Dream as a concept, and then chart a history of its conceptual development independent of the specific uses of the phrase. 

The origin of the phrase is credited to James Truslow Adams' 1931 book Epic of America. One way to use "big data" to look at the frequency of use of the phrase across time is by Google's Ngram viewer. The phrase appears to increase dramatically in frequency of usage starting in 1949, peaks in 1971, declines in the 1980s, has a second peak in 1994, and then tapers off in frequency of appearance. To trace the frequency of use in literary texts, one could do a narrower analysis of digitized texts by separating out ones you consider literary or doing a full text search in the MLA International Bibliography to find its use in literary criticism. 

A conceptual analysis would involve thinking about what American literary works tell a story of "rags to riches" success through hard work. This might include works of Ben Franklin, Fitzgerald's The Great Gatsby, and later works which address the issue satirically such as Albee's The American Dream and Miller's Death of a Salesman.