The first clue to what Fitzgerald is saying is to examine what the main characters consider as important. These items of importance could be seen as "The American dream" for each character. What is interesting is that each character has a unique wish or dream.
For Jay Gatsby, it is to win the love and acceptance of Daisy Fay Buchanan. To do this, he acquires extreme wealth by ill means. Whatever he does is in pursuit of his dream and therefore acceptable.
In terms of the American dream then, one might interpret this as a statement regarding the corrupting power of the dream. Whatever is required to achieve the American dream is acceptable, regardless of its morality or lack thereof.
As for Daisy herself, she can be seen as the representation of the dream's achievement. Daisy has all the conventional American dream has to offer. She has wealth, social connections, and complete freedom to do whatever she wants. Yet, she is desperately unhappy and bored.
Whereas Gatsby is the representation of pursuing the American dream with whatever means available in an almost feverish way, Daisy represents what happens when the dream is finally achieved. She is bored and unhappy.
These two characters can be considered to represent Fitzgerald's views about the American dream. In its most common form, the American dream is a corruptible power. Its projected value is far overestimated, and by itself, it cannot make the dreamer happy. In other words, the American dream is essentially an empty pursuit that ultimately fails to create the happiness the dreamer expects.