Mulvihill doesn't seem like the sharpest tool in the shed, but he's certainly the meanest. He's familiar with Jake Gittes, as they're both rival P.I.'s. The water company hires Mulvihill to guard their reservoir, ostensibly to prevent farmers from trying to attack it and blow it up.
But they actually want Mulvihill to prevent people from discovering that they're draining water from their own reservoir, as part of Noah Cross's plot to steal it.
When Jake sneaks onto the reservoir, and discovers this very thing, Mulvihill and a hired goon (who Jake calls a "midget") grab him. They demonstrate just how serious they are when the goon slices Jake's nostril open with a knife. But Jake gets Claude back, beating him up outside the retirement home that figures into Noah's water-stealing plot.
The moral of this story? Don't mess around with Jake's precious honker.
Later, when Jake confronts Noah Cross about Hollis's murder and the incest, Mulvihill turns out to be working directly for Cross. He pulls a gun on Jake, forcing Jake to hand over the bifocals that Noah dropped when he murdered Hollis. He also forces Jake to take Noah to Chinatown, which leads to Evelyn's death and Noah's successful recapture of his daughter/granddaughter.
Nice guys finish last, and bad guys finish first—and they drag brutal dopes with knives, like Claude and his pal, to the finish line with them.