The character's name was chosen from Emporer Constantine of Rome who first legalized Christianity. Constantine dreamed that God commanded him to pain crosses on all of his warriors' shields for when they marched into battle. Constantine did this and enjoyed an epic victory. Christianity, which at that point had been outlawed, got it's first (real) chance to flourish. Given that the Vertigo comic book character John Constantine defends the world against divine (heaven, hell, and otherwise) encroachment and allows people to live their lives in peace, one might say he was aptly titled.
The movie "Constantine" was based on the British comic "Hellblazer" written first by John Delaney then Garth Ennis. Keanu Reeves played a decent, albeit angstier John Constantine than the basis character -who was a smarmy, smart-aleck. The story worked pretty well on its own with moderately interesting characters. It was a fairly creepy Exorcism-superhero story.
I first saw the movie when I was an extremely pious Catholic.
And I loved the hell out of it.
If you understand Catholicism as a fear-based faith (seriously, we/they pray to God to get them into Heaven because they're afraid of Hell), then Constantine follows naturally. If you're not down for that, it's a playful stab at how the God/Satan dichotomy played out and what's really at stake in the Divine Game.
Like any piece of fiction, it's only as corrosive or offensive as the reader/viewer chooses to make it.