Given his dealings with heaven and hell, should the pre-52 Constantine have ever died what would have happened to him? When he tricked the First into curing his cancer he was destined for hell.
Constantine usually means well and has spent his life fighting evil and saving the world. On the other as he himself admits he is a ba#@%. He knows how dangerous his life is and the games he plays yet often shows little hesitation or remorse to call upon old friends and sacrifice others in his schemes. He involves himself just as often because he is bored or to prove how much smarter he is than someone else. I suppose worse for him would be his constant giving heaven the finger and the corruption of the archangel Gabriel. Heaven would almost certainly kick him out for that and hell would be all too welcoming. I know The First of the Fallen is generally an idiot, but when Constantine died the rules that prevent the First from directly harming him would be removed. Despite Constantine's claims that he would be running hell in a few centuries the First (and nearly every other high-ranking demon) would have to be a complete idiot to listen to him or give him a chance at all to speak.
If you follow the idea that a person goes where they think they ought to go I'm not sure that would benefit John either. Generally, he has shown he feels some guilt about all the people who get involved with his schemes and die because of it. That alone would send him to hell. About the only other option would be to try and bargain with an old god to get in their afterlife or find a way to immediately reincarnate himself.