In the book "the great gatsby": what does gatsby's friendship with meyer wolfsheim imply about his own background?

The character of Wolfsheim is more of a caricature. Anti-Semitism was common in Fitzgerald's time, and it was not considered rude or crude to be open about it. Wolfsheim is based on the notorious gambling figure Arnold Rothstein (who may indeed have masterminded the fixing of the 1920 World Series). Fitzgerald just made Wolfsheim more of a stereotype, with his caricature of an accent ("gonnegtions") and other mannerisms. 
Wolfsheim is presumed to have achieved his brand of success in the same way as Gatsby - using illegal means to acquire enough money to buy his way into the power class. The general theme of the book, in fact, asks that very question - can you buy class? 
Fitzgerald saw Wolfsheim and Gatsby as examples of those who use America's opportunity-based society as a way to rise quickly from poverty and obscurity into the same social circles as the families of old, inherited wealth.