I really hate the backstory they gave him here. Constantine's problems have always been his own fault and no one else's. Arguably, he may have been born with strong magic, but it was still a decision, a conscious decision that didn't involve killing himself.
I really liked a lot of the details, even if they didn't really match comic book world. It gave the whole thing a richness and depth that I have a weakness for. The whole "sending demons back to hell" thing was kind of stupid though. Constantine's priority is himself, always, and he'll align himself with whichever side he feels suits his needs.
OMG THEY KILLED CHAS. Dude, the one rule of the comics is that you don't kill Chas. Kill off everyone else, but don't kill off Chas.
I don't like that they took away Contantine's gallows humor. There are still edges of it left, but Keanu just can't sell it right.He's far too dour and serious. It makes me sad.
I think one of the things they half-got at moments was that Constantine is mostly about bullshitting other people into doing what they want you to. There was Balthasar and Lucifer, but that was all. He was much more about being badass and cranky here, though that can be appealing in its own way.
I rather liked Angela. She's a cool character. The not-romance worked for me.
Overall, I think they got him about half-right. It was entertaining, and there were some good moments, and if you pretend it's not supposed to be Hellblazer at all, it's actually rather good.
I think I have more rambly thoughts about this, but I'm forgetting them right now.