Le Samourai (spoiler)

Saw this famous French film today. Has anyone else seen it?

On the whole I liked it; a fine ,poker-faced performance by Alan Delon,some very moody and stylish cinematography with mathcing music, and a tense ,economical screenplay.

But in the best traditions of film-noir (eg The Big Sleep), the plot was rather confusing especially at the end.

Could someone explain who exactly ordered the first hit? Was it someone else at the club. And why do they order a hit at the pianist.? Why does Delon kill the man at the end? Just revenge?