How did Mr. Gatsby become so rich after being a sailor?

While sailing with his mentor Dan Cody, Gatsby learned to speak and act like a well educated gentleman. This allowed him to keep a reputation as a member of high society and reach people who would not have normally had access to.

When Cody died he had anticipated an inheritance but this turned out not to be.

With his ability to make friends all over the city he became immensely valuable to the criminal underworld who used him to sell bootlegged alcohol (effectively smuggling in the same way a drug smuggler/dealer works today).

With this money he bought a big house, hosted fantastic parties all with the goal of reconnecting with his long lost love Daisy Fay.