This expression is from the show "True Detective" (Session 1 Episode 6) A guy buys a woman a drink when they have just met, then she asks waiter to "martini, extra dirty". This is the first time I've heard this expression. How rude is this? Is this just from Texas and other southern states? I think "extra dirty" means to put some salt and oil in it but I'm not sure.
Dirty refers to adding olive brine to a martini.
Extra dirty refers to adding more olive brine to a martini (approx 2 parts vodka/gin to 1 part olive brine), and usually extra olives too.
See this article for more on dirty and extra dirty martinis.
I would presume that she considers adding all this extra olive flavour to be messing up a martini, or fucking it (as a shortening of "fucking up" or "fucking with").
Hence, she is asking for a martini that has been adulterated, or debased, by the addition of a large proportion of olive brine, leaving it "extra dirty".