Overkill is a feature that reduces the hits done in close combat if one side is vastly outnumbered. The idea behind it is that 100 men can't beat a single victim up more effective than 10 men could. Due to simple physics and crowding, you can only deal so many hits on a small group of enemies. More importantly: The more attackers crowd around, the less effective they will be (stepping on each others toes, being in each others way).
All is not bright for the defenders, though. A unit that was "overkilled", so to speak, will suffer a permanent (well, for the remainder of the battle) increased chance to flee in panic. So even if they survive the close combat, for example, by more reinforcements from the own side joining the following round, they will be less effective.
How to Fight with Overkill
To be most effective, you need to take overkill into account when setting up a battle. Before the introduction of overkill, the most effective tactic was to hit the enemy with as much power as possible, in one large blob of units. That is no longer true. Remember that the more overkill you do, the higher the reduction of hits caused will be. So too much overkill is bad. Your target should be to just reach overkill. By doing so, you will inflict the panic penalty on the enemy units, while wasting only a few hits.
So one approach that will work well is to attack in waves. Instead of one large blob of units, have them set up in waves just large enough to reach overkill. That means your frontline needs to be the strongest, and the lines after it should be just strong enough to replace the casualties. Of course, if you are overkilling your enemies you almost never lose anyway.
Cavalry also work differently now. Earlier, it was best to have them hit together with everyone else. But due to their charge bonus, cavalry forces reach overkill easier than anyone else. So now one method to employ would be to have the cavalry hit one turn earlier then the main force, hoping for overkill which will soften up the enemy and incur a panic penalty (or even cause them to flee outright). Then the main force joins the close combat, saving the cavalry (who don't have a charge bonus anymore) and again ensuring that overkill is reached.
When you attack a fortified city or stronghold, overkill is your friend. The higher your overkill factor, the lower your chances of being thrown back. So if you want to take the place, sacrificing a bunch of soldiers but getting across the walls is the way to go. Attack en masse and overwhelm the defenses. You will score only a few hits, but you will be inside and can take the castle without having to mess with the strong outer wall. Of course, if you have enough siege engines, and don't want to lose too many troops, then you might want to fight as outlined above instead.
Fighting Superior Enemies
All the above applies when you have the superior force. If you face an enemy who is stronger, you will want to use overkill to your advantage in a different way, essentially by forcing them to do all the things which you would try to avoid if you were in their place. Once again, staggering your forces, ideally having one small unit in each combat line, is the way to go. It will ensure maximum overkill and the largest number of wasted hits. Mixed Infantry or even Archers are very good overkill "victims", as they can shoot arrows at the enemy 2nd and 3rd rows while the close combat takes place in front of them, and then shoot at the charging units just before being forced into close combat themselves. That way, an MI or Archer unit should be able to cause twice the casualties it takes.
What you will want to do as the outnumbered party is to cause maximum damage to the enemy. That means lasting as long as possible. Each combat round means equipment and morale damage for your enemies.