The quest ask you to kill a bandit leader in a tower close to Whiterun, but the administrator (Provencio Avenicci) gave me previously a quest to kill him. So that enemy is already dead and I can't do the quest for the Companions. The Companions quest appears to me in "complete quest" side...

Try reloading to before you got the quest from the Companions. It might send you somewhere else. If it doesn't work, then you'll just have to wait for that area to respawn.