The feat Dirty Fighting (originally Sword and Fist 6 but, later, Dragon #303 28, and free here) as its benefit says, "As a full attack action, you may make a single melee attack that deals an additional +1d4 points of damage." Thus the benefit of Dirty Fighting limits its use to a lone attack taken as a full attack action; there's no way to mix the benefit of Dirty Fighting with flurry of blows as flurry of blows also takes a full attack action. One must pick either to fight dirty or to flurry.
That's okay, though, because the feat is terrible. Attempts were made to rehabilitate the feat by making it a short feat chain with the feat Improved Dirty Fighting (Dragon #303 30) or a different version of Improved Dirty Fighting (SF errata 2-3); neither of those feats works with flurry of blows, either, both still requiring a full attack action.
I understand that the goal might be something role-playing appropriate—and I totally understand that a drunken gnome fighting dirty could be hilarious—, but that can instead be the player describing his character's actions. You don't need to take a feat to say you headbutt the bad guy in the groin, and no one should take the feat Dirty Fighting.