How would I begin writing a literary analysis paper about Jay Gatsby?

To write a literary analysis paper about the character Jay Gatsby, you could go in several directions. First, you could ask what Fitzgerald, through Nick, is trying to say about Gatsby. What does Gatsby represent? On one level, he's just a criminal and grifter, making a vast amount of money by cheating people and selling alcohol illegally—and he's trying to steal another man's wife. Nevertheless, to Nick, he is a positive figure. To Nick, he represents the American Dream. How is this symbolized? Could it be, in part, through the color green? Where in the novel is Gatsby connected to the color green? Why is Gatsby always also connected to the color yellow/gold and why?  Could one color connect him to the idea of having a dream or desire? Could the other one link him to wealth? How are these two elements of the American dream (wanting something badly and money) portrayed through Gatsby?
Another route would be comparison/contrast. What traits in Gatsby's character become clearer when we compare him to Tom Buchanan? More specifically, how is it that Nick is always miserable around Tom? Why does he usually have a good time with Gatsby? Does Gatsby have a friendlier, kinder personality that focuses on other people and their needs? What places show this? Think about his interactions with Nick and how Nick feels about them. Think too about what Gatsby does when someone's dress is ruined at one of his parties. What does this show about him? What might Fitzgerald be saying is important in life? Is it coming from old money like Tom, with a string of polo ponies and racist theories about the superiority of "Nordics," or is it behaving with some element of kindness and generosity?
Best of luck, whichever way you might go!