I second what everyone above me has said, Injustice is a very inaccurate depiction of Superman's character.
I'd recommend checking out Action Comics #796. Despite how old the story is, I think it's worth a read and I don't want to spoil it, but just to explain why it's relevant to this discussion, Manchester Black makes Superman believe he killed Lois, and despite Superman's temptation to act in violence, and Black continuing to try to provoke Superman, he does not attack Black, because vengeance is not justice, and he will not break. The story really holds up and I think it does a great job showing Superman's mental resilience and morals.