I think the major theme you need to analyse and relate the character of Jay Gatsy to is the American Dream. Some have called Jay Gatsby the most intriguing figure in American literary history. He is a character who has literally formed his own identity and embodies in so many ways the American dream.
Consider how his story emerges and develops this theme of being a self-made individual. He comes from humble roots as his parents were poor farmers. The Gatsby that we meet in the novel is a result of his single-minded pursuit of the American Dream and his desire to marry Daisy, who, for him, personified that dream. Daisy becomes his sole reason for being, and thus he sets himself on his path of, as it is described, "following the grail" so to gain himself the social standing and wealth that would place him in the same sphere as Daisy and give him the chance of winning her. This leads to his involvement in a drug ring, allowing him to gain wealth and therefore buy a house close to Daisy's and throw luxurious parties in the hope that Daisy will attend.
It is clear that as Nick gets to know Gatsby and plays the role of observer and actor in the events in the novel, Gatsby is presented as being insecure in his position and his parties turn into gratuitous events that are peopled by disinterested and self-absorbed socialites. However, Gatbsy's profound loyalty to the American dream and his idealism distinguish him as being a real tragic hero in American literature whose death is in part a result of his single-minded pursuit of the American dream.
So, when thinking about the character of Gatsby we cannot help but relate it to what Fitzgerald is trying to say about the American Dream as a theme in the novel.