In Recettear, I'm slightly confused about how building up a "chain" of monster kills works. At first I thought it was determined by the amount of time I take in between each kill, but this doesn't seem to actually matter. It doesn't seem to be based on whether or not I take damage, either. How do I increase my kill chain? Are there even any rewards for increasing it or is this mechanic just for aesthetics?
Chains occur by killing multiple monsters of the same type. You start a chain of 1 once you kill a second monster of the same type. This causes you to get more experience with every such kill. The same type means the same color and kind - blue Slimes are different from green Slimes, but both kinds of red Mushrooms will chain, for example.
For example, suppose you run into a room and there are four blue Slimes and three blue Eyebats. You could kill them in any order, but if you kill all of the Slimes first then all of the Eyebats (or vice versa), you'll get more experience due to chaining. Chaining does not time out, but it will reset whenever you kill a new kind of creature.
In general, unless you've got adjacent rooms with the same creature type, you mostly want to ensure chaining within each room segment and not go out of your way to chain. You'll get more than enough experience in this fashion.
There are some exceptions to enemy chaining. These exceptions all are treated as the same as one other kind of enemy, but they cannot increment the chain count. They will cancel any chain that is not of their kind. Note that while killing them can never increment your chain count for that enemy type, it will count as the first kill towards starting a chain. As such, you will want to kill one of these non-chaining enemies before killing the real enemies in order to maximize your chain potential, leaving the remaining to take advantage of your maxed chain.