Billy Jack can be found in one of the bars in New Orleans, where he can be seen talking to a fellow crow member and yellow guard bird.
He seems to be working with the group headed by Mark Purayah II, and in conversations that take place, he is ordered to deliver a briefcase full of diamonds that is payment for the execution order on Jimmy Cilley. His successful delivery of the suitcase is tantamount to the mission's success. If 47 intercepts the payment before it's delivered, 47 basically has unlimited time to take out his three targets without having to worry about Purayah II giving the kill order, and 47 can exit with the payment briefcase in his possession to add a bonus to his payment for the mission.
He wears a red bird costume and can be seen pausing to smoke several times in the game.
He can almost always be seen in the blues bar in New Orleans, or later in the game at The Crows' hideout.