Martini, Extra Dirty

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".