Jay Gatsby is the character in the novel around whom all the action revolves. As such, he is the protagonist. Nick Carraway shares Jay's story with the readers who promptly know that the story being told is about him, since the title of the novel names him directly and the following extract, immediately after the introduction, further emphasizes the fact:
Only Gatsby, the man who gives his name to this book, was exempt from my reaction — Gatsby, who represented everything for which I have an unaffected scorn.
Nick, in the role of narrator, informs the audience about Jay's dream, his holy grail, which is to win back Daisy Buchanan's affections and recreate the past. He informs us about the extreme measures Jay takes to realize his ideal. We learn about his new-found wealth and his relationship with shady characters, such as Meyer Wolfsheim, from the underworld. It is as if Jay is some fantasy character seeking to fulfill an impossible dream.
Nick expresses both scorn an admiration for Jay - firstly, because he represents everything about society which he despises: its greed and accompanying ostentatious materialism and secondly, its infinite hope, aptly symbolized by our protagonist in his quest for the realization of an ideal. It is this desire that, ironically, culminates in Jay's destruction and brings to the fore just how empty and meaningless the extraneous quest for happiness is.
Jay's poignantly tragic tale serves as an admonition that the desire for satisfaction, peace and happiness cannot be found outside oneself. Jay never achieved what he desired, for he sought, as Daisy brutally told him, 'too much.' Unfortunately for Jay, he did not want to believe in reality and tried to find happiness in the past and that, in itself, made his quest a piteously futile one.
The question of whether Jay Gatsby was a bootlegger resulted from the many rumours that were bandied around about him. His past was mired in secrecy. The few glimpses we have into his history informs of one who dreamed of a bright future and who was prepared to work towards it, as illustrated by the careful notes he kept as a young man. Central to that future was reuniting with Daisy Buchanan and Jay did everything possible to achieve that ideal. He had no education or the background to provide him with a solid foundation, which obviously put him at a great disadvantage.
When Jay was introduced to the taste of wealth and the privileges it afforded when he met Dan Cody, he must have believed that the only manner in which he could possibly achieve his dream was by taking that route. It is then that he decided to achieve his dream by indulging in illegal activity - a risky shortcut to wealth. He formed associations with members of the underworld and achieved enormous riches within a few years.
So, yes, Jay Gatsby must have been a bootlegger, among many of the other criminal activities he felt obliged to indulge in. There is no other way to explain his achieving so much wealth in such a short period of time.
Gatsby's role in the novel is to illuminate the life of the wealthy in the 1920's and the extremes which some took to achieve the American Dream. His ladder climbing had a double-edged sword since he only wanted wealth to impress Daisy and to win her love. However, for all her shallowness and "full of money" voice, she could not love someone whose wealth was achieved in a less-than-honorable profession.
While we do not know for sure that Gatsby gained his weath through the illegal activity of bootlegging alchohol, it is certain that his dealings were somewhat shady since he was in close contact with Wolfsheim. The air of mystery that surrounds Gatsby from the beginning of the novel (rumors of his having killed a man, or being an Oxford man, or spying in the war) doesn't help his case for being a man of honorable means.
His death is fitting then, since he uses illegal means to gain his wealth, but in doing so he loses Daisy. He is killed as a matter of misunderstanding (Tom intentionally misleads Wilson to Gatsby) thus causing even more mystery and shadow to be cast on him as a member of the nouveau riche. This sort of dishonor could never touch Tom from the old-money bunch.