In some of the dialogue, Columbo refers to the fact that he killed someone so as to avoid getting found out. Columbo says that he felt sorry for Pete, seeing him as a decent person who got in over his head. Nevertheless, the cover-up murder would be what involved Columbo.

I have only seen this episode in its uncut ABC version. Maybe this explanatory dialogue was edited out in the version you watched.

The main hatred Vivian Dimitri had with Columbo is that he hounded her late husband and also eventually arrested him. Regardless of whether or not he was guilty of murder, Vivian loved Pete and this drew hatred for the people involved in his demise. If succeeding in killing Columbo and his wife, her next step might have been going after the judge who presided at Pete's murder trial.

By the way, I am glad that the Pete case referred to in "RIP Mrs. Columbo" was not an actual case from a previous episode. It is fortunate that the fate of villains from previous episodes were allowed to remain up to the viewer's imagination.