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My personal speciality it's not required in most public games, but it's a fun thing to do. Hunting uses most of the skills previously covered. It only happens when you are in the dominant team and there are weak or few enemy's left somewhere on the map. Hunting any other time will get yourself shot.
The main battle and fire fights have died, so all is left is ambient map noise. First thing, stop running and slow to a walk, there is no rush. Move across a broad area of the map, trying to cover the most areas. Pay attention to left overs from battles, like guns. Are they your teams, or the other teams, which way are they laying?
Don't make a single noise, and listen out for a single step of a cowering bad guy. When you hear that single step, follow to where it was, and explore. Check EVERY corner you pass, but do not fire until you need to. Do not get into a pattern and move around the map in a predictable way. Double back on yourself from time to time, and take odd routes. But make sure you cover as much ground as possible - check all the logical places first, the enemy's side of the choke points, dark corners, sniper spots etc.
If you meet up with another player on your team you can change your style of hunting. There is no longer the need for stealth, as having two of you makes you a force to reckon with. Run side by side, checking your side's corners and camping spots. Stick together and again try to cover as much as possible.
Hunting alone is a stealthy slow thing, hunting in groups is a noisy show of strength. Of course you can be noisy when alone and quiet when in groups, but I find this is the most effective. It all depends on your opposition. Before you start your hunt check out your opposition, do they have more kills than deaths? What's their ping like? Do you remember seeing them hide somewhere before? All these can give you an advantage, and help you decide how to attack them. You may find that the last man standing is the best player on the team and is luring you into traps - you may want to be very careful and not all get caught in the same trap.