According to several different Latin dictionaries, including Lewis & Short, the word peremo (to destroy) has the etymology of being the combination of PER + EMO. However, I thought emo meant to buy or purchase, so I don't see how this is a root word for peremo which seems to mean something totally different.
While emō normally means "buy", the ancestral meaning seems to have been something like "take". Compare the Proto-Slavic cognate *ę-ti "take, acquire", or the distant Latin relative pōmus "fruit" (the first part is cognate with Greek apo: fruit is "something taken off/away from" the tree). And indeed, this is the meaning we see in most prefixed forms: adimō "deprive", dēmō "remove".
So the original meaning of perimō seems to have been something like "take away thoroughly and completely"—that is, "cause to vanish". And indeed, in Classical times this still seems to be its most common meaning: Cicero in De Divinatione, for example, describes how the full moon was subitō perempta est (i.e. eclipsed). But from there, "cause to vanish" poetically meaning "kill, destroy" is an easy step.