Brothers and Sisters: The Dad dies to start it off.
Six Feet Under: The Dad dies to start it off. And then someone new dies at the beginning of every episode.
Desperate Housewives: The neighbor dies to start it off, and then becomes the narrator.
The Shield: An undercover Internal Affairs investigator is murdered in cold blood by the protagonist in the first episode, setting up the (a)moral context for the show.
Party of Five (?): I know the premise is the dead parents, but do they actually die to start the show?