Honestly, Constantine's magic is even less Thaumaturgy then just straight up plot device - it'd probably be handled much better with invokes and declarations than Thaumaturgy, I think.
I'd say he's probably got the Ritual power for exorcisms (he seems to have legitimate power and skill there), and the rest of what he uses are basically common ritual type stuff -- he bargains with some other power, or uses some ready-made item.
So he'd have a very high Lore score, with some stunts to boost it for diabolism and possessing demons. Probably also a stunt to represent Zed's dad's place as his workspace (replacing Resources with Lore, maybe?).
Possibly he cashes in fate points and refresh often to "borrow" the items of power in Zed's dad's place.
If you're talking about bringing the character into DFRPG...I'd honestly make him a Pure Mortal instead of giving him a template. Leaves him a ton of Fate Points to invoke for effect his aspects all the time for those types of Plot Device things - his magic seems less like what a practicioner does, and more like something anyone could do if they knew the words. Constantine doesn't seem much more of a practioner, with innate magical talent, than Thomas does - he's just go really high Lore and crazy Aspects.
I don't see him as the type of character who starts with a ton of fate points -- he's
definitely someone who gets compelled left and right, builds up his fate points that way, then cashes them in right at the end for some insane and improbable resolution.
Like this most recent episode, I could see his player going, "I'm going to have a smoke, call it a self-compel that Pazuzu gets to do whatever it wants in the next hour or so in-game."
But that's just part of the universe he comes from. It doesn't seem like the spells he casts come from him, but that, for instance, he could instruct pretty much anyone to do any of his spells (clearly, Dresden couldn't tell Murphy "Okay, just hold this item and say these words" and expect to get some Thaumaturgy out of it).
Actually, he could, with common ritual.
So, overall...I'd say Ritual (Exorcism), with Lore as his top stat, Endurance equal or just below, with various stunts using Lore for things like Contacts, Resources, and such, and adding a mild physical consequence (he can't fight very well, but he can take a punch like nobody's business). Definitely the Infuriate stunt; maybe an Aspect boosting it, even.