Ward Fowler, who plays the popular TV detective Inspector Lucerne, kills his agent because she is blackmailing him with the fact he was a draft dodger. He drugs his alcoholic studio gopher while they're watching a sports broadcast, tapes it, then goes to the deli where he knows his agent will be. Disguised as a thief with clothing from the studio wardrobe department, he pistol whips the owner, then shoots the agent and makes it look like a robbery. He then returns to his home, wakes up the gopher after resetting his watch, and plays back the tape so he has an alibi.
Columbo is on the scene. A big fan of Fowler's show, he immediately begins to notice certain discrepancies about the crime scene. Claire's purse and credit card were left behind, and because of the difference between the bullet hole in her shirt and in her back it looks as if she was shot with her hands up even though the story seems to support her turning around to run away (when her hands were down). But Fowler has learnt a trick or two by playing a detective, and leads Columbo to suspect the agent's husband. He even plants a thread from the husband's sweater on the gun that he borrowed from the prop department.
Seeing past that, Columbo notices other clues. The gopher's expensive watch is in fact not set correctly (he always keeps it five minutes ahead). And the agent was shot straight through the heart from 30' - Fowler was a sharpshooter in the army.
Finally Columbo puts it all together and comes up with Fowler's one mistake: although he cleaned the prop gun thoroughly after he borrowed it, he forgot to clean the blanks. Fowler's fingerprints are all over the blanks, incriminating him.