If the mockingbird symbolizes people who have done nothing wrong but are hurt, how might the idea that it is a sin to kill a mockingbird apply to the novel (Part one of the book!)

First, let's rephrase the premise put forth in this question: Atticus explains to Scout that it is a sin to hurt a mockingbird, and Miss Maudie clarifies his point when she says that mockingbirds don't do anything except make music and sing. Miss Maudie's point about the mockingbird emphasizes the innocence of the creature, who is undeserving of pain and punishment.
This point applies to the novel as a whole when the reader takes into consideration innocent victims like Boo Radley and Tom Robinson.
Boo is a recluse who wants to be left alone to his shy, solitary ways, but the community's gossipers torment him with their expectations of social behavior. Tom is a gentle and open-minded black man in a southern community where most white people see him as a lesser citizen for no reason other than his appearance, which is out of his control.
Like the mockingbird, both men are innocent. They are harassed and forced to suffer at the hands of others who are more powerful than them. Atticus would most certainly describe this kind of treatment as sinful.