Edwin was a character who really grew on me over my first watch of the series, and especially over the first episode. Initially he comes across as aloof and distant with everyone but Charles - his cool demeanour with the WWI nurse and "playing hardball" with Emma, his obvious resentment of Crystal's intrusion into his space and his irritation with Charles for inviting her without asking him.
One of my favourite little character-revealing exchanges in the first episode is Edwin and Charles' argument in the games cupboard about whether to take the Becky Aspen case - specifically, the way that after Charles lays out all his arguments about taking a break from the office to avoid getting caught by death, he smiles, relaxed and confident, and pulls out what is obviously his trump card:
"You really gonna let a little American girl die?"
And Edwin gives him a longsuffering look, and Charles grins like a madman, and in the next shot they're in Port Townsend. Because no, Edwin was never going to do that, and they both knew it.
We see more of why he's like that over the rest of the episode, the flashback to his own murder, and then finally in his argument with Crystal, where that composed facade cracks and we see the thing that drives him, the reason why he's spent the last 34 years risking getting caught and sent back to hell, the motivation that normally hides behind the sterile excuse that a good detective does what he must to close the case.
Edwin wants justice. For himself, for Charles, for Becky Aspen, and yes, for Crystal too. He wants every death to count, every hurt to matter, every wrong to be righted - or at least acknowledged and given the weight it deserves. He was murdered at sixteen in a hate crime, was sent to hell on a technicality, and dragged himself out to discover that his death had been smoothed over and forgotten. And he's spent the last 34 years doing everything he can, even at the risk of being sent back, to make sure that kind of thing doesn't happen to anyone else.