I got into this series after going through 6 compiled editions of Alan Moore's Saga Of The Swamp Thing. John Constantine is introduced in that series as a smirking, supercilious mystery-man who leads Swampy on a series of increasingly bizarre encounters with the supernatural world, taunting and playing mind games with him the entire time.
In the Hellblazer series, Constantine is fleshed out a bit more, and shown to be an actual human being, with fear and doubt and sleepless nights spent in the company of ghostly tormentors. I think the writer, Jamie Delano, gives the character a great deal of depth, and smartly uses the plot twists to give shape to his complex psyche.
The other thing that makes these stories interesting is the art style. I find the artwork to be almost impressionistic in its depictions. It gives everything an ambiguous, grainy quality that really underscores the mood. The mood is crucial, because without it, these stories wouldn't have the emotional nuance to carry the demons-and-brimstone theme around without seeming a bit silly.
Each chapter seems to be introduced with some sort of poem, which is a nice little touch.
I'm reading the series in chronological order, and as I'm writing this, I'm embarking on the follow up compilation, The Devil You Know, also written by Delano. It may turn out that other writers of this series are able to accomplish more than what Delano has, but so far, it is a solidly written series, that's very easy to jump right into.