Jay Gatsby does not change much over the course of The Great Gatsby, but what we know about him changes immensely.
The actual changes in Gatsby's character relate to an emerging openness in his feelings. Daisy causes Gatsby to act nervously, impetuously and in effect to lose control of himself. This is a change from the collected and powerful figure he presents on first meeting Nick.
Gatsby grows more desperate as the novel continues, yet his fundamental attributes remain intact. He is self-involved, self-willed, and insists on making his vision come true. He continues to pose as a virtuous man with a moral center, despite the facts that 1) he is trying to steal another man's wife and 2) he is a criminal.
In this sense, Gatsby himself does not change in any significant ways but remains the same poseur he was from the beginning. What does change is our view of his character.
Gatsby is not a confident successful man, but a starry-eyed dreamer willing to trangress traditional moral lines. He is not Jay Gatsby but Jay Gatz. His persona, or mask, is penetrated over the course of the novel, while the man behind it remains largely unchanged.