I know there were previous questions asking whether it is OK to kill a pest, say a fly, but the consequences are unclear. So I'm asking if killing a pest makes a person unclean.
See Leviticus 11:29. Animals that walk on four paws, with their bodies clearly high off the ground (e.g. a dog or cat), have carcasses that would make you "ritually impure." The only smaller animals with this property are the seven "creepy-crawlies" listed there, e.g. a mole or mouse. So if you killed a mole or mouse and then touched it (or touched a dead mouse that died on its own, for that matter), you'd be "ritually impure." (All you'd have to do to fix that status, however, is immerse in a ritual bath and wait until nightfall.)
However this is a moot point today as we assume we're all ritually impure, and we don't deal with laws (e.g. Temple stuff) that require a state of "purity." Contact with a human corpse makes you a lot more "ritually impure" than contact with a dead mouse, and yet those who are involved in Jewish burial societies are held in very high regard. The whole "ritual purity" thing is limited to certain laws. (Note that a kohen is still prohibited from touching a human corpse today, but can touch dead critters all he likes.)
A dead bug, however, wouldn't even cause a state of ritual impurity.