This is an interesting comparison! I think that the biggest way in which Gatsby is like Don Quixote is that they are both eternal optimists. Gatsby believes in "The American Dream" - the idea that he can be whoever he wants to be if he works hard. So his life is spent trying to re-invent himself. He lives with a wealthy benefactor, and when he dies, Gatsby goes into the army, gets out, goes to college, and becomes wealthy albeit through dubious means.
Alonso Quixano is also unhappy with his lot in life, so he, too, creates another persona for himself - that of the knight Don Quixote de la Mancha, pursuing his beautiful Dulcinea. Like Gatsby, people take advantage of Don Quixote, but unlike Gatsby who never really seems to lose his innocence in the novel, Don Quixote seems to grow wiser and cynical towards the end of his life. Both men die in the end, never truly achieving what they set out to achieve.
Read more about it here on eNotes.