So we all agree that River has...(spoilers)

River only sees the 'spirits' of victims until they are avenged (i.e. their murder solved), he keeps seeing "Thomas Cream" (played wonderfully by Eddie Marsan) who taunts him about his (River's) guilt and says that he'll always be there ("I am your hell, I'll never leave you").

There is only one theory that would explain this (without any extra assumptions), so we must all agree that River has killed a man, probably a serial killer who River failed to arrest.

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Fire is my Champion