How is Jay Gatsby motivated by a dream of a more just world?

First, I would refer you to elsewhere in . The world Gatsby lives in is fundamentally amoral. Gatsby himself is an expression of this moral ambiguity. To quote the linked answer:

The character of Gatsby is particularly ambiguous. In his daily actions, he is fair and just. Polite almost to a fault, he is somewhat full of self-doubt. He is polite and thoughtful. His flaw is that he believes an illusion. He dedicates his existence to a return to the past--his relationship with Daisy. But the relationship never really existed as Gatsby experienced it. Daisy never loved him as he loved her. This doesn't make him immoral, however, only deluded and naive.

For Gatsby, a "just world" would be one in which he and Daisy can be together. The big house, the huge parties, are all a means to an end, the end being creating the environment he thinks will bring Daisy back to him. It doesn't matter to him that this is all paid for by illegal activity, or that presumably his criminality has victimized many people. If he can just get Daisy back things will be as they should be. His fixation on reliving the past is an expression of how he confuses the personal and the (for lack of a better word) cosmic. To Gatsby's mind, in getting Daisy, he heals himself, and in so doing, heals the world. Of course, this is not the case. Daisy doesn't love Gatsby the way he loves her; in fact, Gatsby doesn't even really love her, but the idea of her. Of course it is wrong for him to try to break up her marriage with Tom; the fact that Tom is such a terrible person only underlines how rotten everything in Gatsby's world is. The moral hole at the center of Gatsby the novel is too large for Gatsby the character to fill. In fact, we come to realize, Gatsby is that hole.