Apart from the more common slang usage of dime to mean 10 cents, it seems that in some gambling circles a dime is also a $1000.
A vig, or vigorish, is the "tax" taken by a bookie on a straight bet. Say you bet $100 - if the vig is 10% you'd have to stump $110.
Is this "realistic" in a "real world" context, probably not, but there are factors in the movie which help explain what is going on.
Firstly, Klingman is not in fact the owner, he merely is the Gambling Licence holder and is a front for criminal interests.
This is mentioned earlier when Michael meets Johnny Ola.
The hotel's registered ...