Improving your skill with melee weapons and in Hand-to-Hand is a simple matter of hitting something. You get 1 skill increase point (base value) for each successful attack. The exception to this is Short Blade weapons where the skill increase point value is only 0.75. However, lest you think that this is a penalty, the speed factor of short bladed weapons is about twice that of other weapons, so you'll get in more swings in the same amount of time.
If you'd like to increase the skill quickly, use weapons that do less damage. This means more successful hits to kill the same critter. Which means more skill increase points. The downside to the equation is that while it's taking you longer to dispose of the critter, the critter has more time to dispose of you. When you're dealing with creatures that deal out a lot of damage or have attribute-draining attacks, doing more damage is more important than gaining a couple more skill increase points. On the bright side, swapping weapons is instantaneous (unlike Daggerfall where you had to wait for the "equipped" message), so it's no big deal to carry a cheesy weapon for the riff-raff and a real whopper for the heavy-duty bad guys.
You have a game option that automatically uses your weapon's best attack form (Chop/Slash/Thrust). Turning this off (I believe it's off by default) will also result in your having to make more attacks. Turning it on will result in your dealing out more damage to your opponent, so you won't receive quite as much melee damage.