After watching 4 seasons of Columbo on DVD; I think there are six qualities that make a good episode:
1) The murder must be clever and well planned
2) Columbo's solution to crime must be logical, smart, and believable
3) The Villain is sympathetic and/or interesting and well acted.
4) The cat-and-mouse game between the Killer and Columbo must be interesting
5) The other elements of the story (supporting characters, comic relief ) must be done well
6) The direction and script have to be tight - not padded or slack.
I've called the murderers "villains" but most of them aren't. Columbo (and me) like them. Some of them are so sympathetic or interesting its too bad they couldn't be let go.
Probably the most common mistake is having Columbo latch on to the killer for no good reason. In the best episodes, Columbo either has a good reason to suspect the killer at the start or quickly finds evidence that leads him to suspect. Usually a murder has occured and the Killer is an obvious suspect except he has a seemly "airtight" alibi that Columbo has doubts about. Or Columbo decides an "accidental" death is really murder at which time the Killer becomes an obvious suspect. In the poor ones, Columbo simply decides on little or no evidence that the killer with an airtight alibi is guilty. The other mistake is having the killer do something illogical and stupid thereby allowing Columbo to arrest him. In these weak stories, the killer will allow Columbo to solve the crime by trying to kill someone else, plant evidence, or worst of all, try to kill Columbo himself.