How could you do such a thing? It is a rhetorical question, one that does not expect an answer. If could be said as: What you are doing is awful. I wonder how you could even do it at all. It is usually said in response to some bad insult or some awful act against another person. If a man touched a woman without her permission, she might respond with "How dare you!?"

It means that the speaker is objecting to something another person has said. This is something that only occurs in a conversation. If you see it written, it will always be part of verbal conversation in the plot.