It's important to know what you're getting into before you start fighting with another man in the middle of what is a sixty-minute punctuated fight already.
First: punch the man in the helmet with your at-best gloved hand. Everyone knows the head is the most important part of the body, so go right for that. Take your hand, a series of tiny breakable bones held together by spirit gum, and throw it directly to the side of their helmet. It won't break and nothing bad will happen for you at all.
Second: grab the facemask. Control the head and the rest will follow, so follow a strike to the helmet (or perhaps repeated strikes to the side of the helmet) by grabbing the facemask. Drag the person wherever you like: the purpose of the control isn't so much direction as much as it is power, i.e. "we are going to the concessions stand, cornerback who punched me in the face at the snap. We're getting a hot dog and no, no we are not getting candy no matter how much you ask."
Third: go to ground. Get the fight on the ground, because this is really the aim of the fight all along. You want to get on the ground, where the two of you can tussle and pretend to fight because, seconds after starting the fight, you have both realized 'we are playing football, and thus already fighting.' No one has ever won a football fight (although Andre Johnson came close once, because he is Andre Johnson.) Just fall to the ground and take a breather while trying to figure out why you did this in the first place.
This is already a sport based on combat, you say to yourself, sort of rolling on the ground with your broken fingers threaded through a facemask and waiting for your teammates to do their part and theatrically pull you apart. Why on earth would you fight in the middle of it? This is like the time you had a friend who was high as balls say "watch dude," and they put diced onions on their onion rings and nodded like "YEAH" and you thought: but it's already got onions, and you only think this is brilliant because you are high as balls. Football fighting is the dumbest, and don't ever do it.*
*The exception is the Memphis/BYU brawl from the Miami Beach Bowl. That should have happened because it pitted two natural enemies against each other for the first and only time in history: people from Provo, and people from Memphis.
But otherwise, no: don't fight between plays, and just take advantage of football's rules that make assault legal and do it during game play unless you're into redundancies unless you're into redundancies unless you're into redundancies. (P.S. You can also do it if you're LeGarrette Blount because no one's ever stopping you, anyway.)