In this sense, undoing is a noun meaning the cause of someone's ruin, downfall, or disgrace — the thing that destroys them. It lives almost entirely inside one pattern: "X was his undoing" or "X proved to be her undoing", and very often "his own X was his undoing". "His pride was his undoing." "Greed proved to be their undoing." "That one careless email was his undoing." There is nearly always a touch of irony, because the undoing is usually something that belonged to the person and once helped them — their confidence, their charm, their cleverness — which then turned round and finished them. That flavour of a fall you brought on yourself is the heart of the word, and it is why it appears so often in stories, court reports, and sports writing about a great player's last season. It is a formal, slightly literary noun and a fixed phrase; "the undoing of" also works — "ambition was the undoing of the company". Do not confuse it with the ordinary -ing form of the verb undo, which is much more common and quite different: undoing your shoelaces, undoing a button, undoing the damage, undoing years of good work. Same spelling, everyday meaning, no ruin involved.