Jay Gatsby's rise to fame was initiated by his meeting Dan Cody, a wealthy gold miner. Cody became like a mentor to young Jay Gatsby and Gatsby (then Gatz) got a thirst for the good life. After Gatsby joined the army, he met Daisy and he fell in love. Knowing he did not have enough money to ask her to marry him and/or offer her the "good life," Gatsby went overseas, planning to return and make his fortune in order to have the good life with Daisy. However, while he was away, Daisy married Tom Buchanan. Still determined, Gatsby recreated himself and accumulated a decent amount of wealth while living across the water from Daisy, every night dreaming of winning her back. In accumulating all this wealth, Gatsby engaged in illegal activity (bootlegging, drugs, and possibly gambling); mostly in connection with Meyer Wolfsheim. Gatsby's rise to fame in trying to win Daisy back from Tom is also part of his downfall. Engaging in illegal activity was dangerous in and of itself. But getting involved with a married woman (Daisy) was dangerous as well, especially considering how heartless and vengeful Tom Buchanan could be and eventually was.
In the end, Gatsby's pursuit of this American Dream (wealth and winning the girl of his dreams) was doomed from the start; and this says something about the growing myth and or elusiveness of the American Dream. Since Gatsby had no money, he couldn't marry Daisy. By the time he accumulated money, she was already married. By the time he made his move, she was married to a man who was also having an affair. With all of this cheating and deception going on, something would eventually break and it did.
Another large part of Gatsby's downfall was his perception of Daisy and his conflation of her with money. He associated wealth and Daisy together as the good life. He even remarked at one point that "Her voice is full of money." The conflation of money and love is problematic. In that sense, Gatsby's rise to fame was also his downfall because he was pursuing too much; an impossible ideal. The dream, as wonderful as it was, could not work in reality. That's why this notion of the American Dream is often discussed as elusive or a myth; as a "dream" it works, but when manifested in reality, it is sometimes or often met with struggle and disaster. In Gatsby's case, his rise to fame was also his downfall because both included engaging in illegal money-making schemes, superficial social parties, and finally extramarital affairs compounded on other extramarital affairs. In this sense, Gatsby's rise and fall were interwoven and could even be considered the same.