There are as many right ways to do this as there are campaigns.
For one thing, the New World of Darkness was designed without so much of an over-riding myth-arch.
The factions are weaker and have less overall control over the world of mortals.
The past is less well known in many cases.
Faction rivalries are no longer consistent. In fact, most factions have their own troubles and rarely deal with each other.
The doom of a soon-to-come apocalypse is no longer a given.
The idea of pervasive moral decay is optional.
A lot of people thinks this gives the game less bite than Classic World of Darkness.
However, the grimdark nature of CWoD is still easily doable with NWoD.
All that above is fluff, and easily changeable.
The first question is what sort of theme do you want for your game.
If you want an out and out horror theme, then you might want to cut closer to CWoD. Horror is not about Good and Evil. It's about how much knowledge your characters have of the way the world actually works. Take a look at the Chinese horror movie, the Eye, the original, not the American adaptation. There is not one single evil entity in that story, there is one hostile entity, but I doubt that one was truly evil. Despite a lack of anything evil, the story remains one of the most terrifying movies I've seen. Elevator scene...gah! Anyway, the whole point of the terror there is that the character is faced with knowledge of something that she can't do much of anything about. (The American version ruins this by making what were silent dark figures escorting dead souls onward into snarling figures snapping at people interfering with them...us Americans, have to have our discernible bad guys...rolls eyes)
For another comparison look to Cthulhu and Morgoth or even Sauron. In the beginning, before his power begins to wane, Morgoth's power was easily superior to that of most of Lovecraft's critters, and Sauron was likely more dangerous personally than Cthulhu specifically. And yet, the setting they're part of is fantasy, not horror. The reason for this is because the natures of Morgoth and Sauron are well known within the world setting, at least by the educated. Their existence is considered common knowledge. (that said, direct observation by the Eye of Sauron still drives things mad....and even the Nazghul have effects on sanity nearby similar to that caused by some of Lovecraft's entities.) Meanwhile, Cthulhu is not well known and considered superstition by most who hear of it.
For a more direct comparison, the Deep Ones in Lovecraft compared to the Sahaugin in D&D. Similar designs (I think the Sahaugin are based on the Deep Ones, actually) and cultures. Now, when a character in Lovecraft's universe stumbles onto the Deep Ones, he's not likely armed for conflict and finds a hidden community. The community hunts him down to keep their secret. So he has to deal with the mental stress of combat and running for his life while woefully and piled on top of that is the discovery that "hey, these people aren't exactly human". The inhuman discovery is more the straw on the camel's back than the cause of the break....the trauma of running for your life and being hunted by a mob is enough to cause mental distress to lots of people. On the other hand, Sahaugin are known to exist in D&D areas and in those places, people consider it foolish to travel without some means of defense. So the stress of danger and the stress of discovery are both lessened.
Now, in a multifaceted campaign, which I'd love to be in, actually, you can arrange horror by starting with the characters as raw neophytes in their factions being thrown into the deep end.
As an adventure, you can have a clear bad guy and objective for the characters to accomplish, but put twists here and there.
More likely, NWoD will cut towards the intrigue and espionage storyline, which is the so-called "grey vs grey" morality. Note, you can still have "good guys" in a greyvsgrey setting. I tend to subscribe to the Granny Weatherwax idea: "There's no grey, there's just black and white that's not so clean." Or also the idea of "I may stand in the darkness, but I can look to the light".
As to fitting the characters together...the facts remain the same as for any campaign:
The characters need a reason to work together.
As such, a collective goal, or entangled goals, are needed. Otherwise it will be hard to maintain the campaign.
That said, there are several options:
perhaps you can put the city they're in under some kind of siege. People going missing (mortals and supers), spirit activity increasing, etc. Enough so that the local factions have moved from "stay out of our way, we'll stay out of yours" that they usually have into a full official truce/non-aggression pact. Now the source of the siege is up in the air.
Maybe some Pure shamans are riling up the spirits to soften the city. Maybe it's the servants of the Exarchs, or some rogue vampires. Perhaps one of the Hunter Conspiracies is running an experiment or an op. Maybe someone's pulling the strings on all of them.
Also note that each of the characters could act as representatives of their various factions. This would be analogous to something like a Vampire campaign where the party would let the Nos talk for them among Nosferatu and the Ventrue among Ventrue. In this case, you'd have the mage speak with the mages, the hunter speak with other hunters and so forth.
Even more so, however, since in a single hunter campaign there would be a status for status among all vamps, not just one group within the vamps. There isn't really a set status merit for status among all supernaturals.
There isn't a Gandalf character whose recognized by all the Free Peoples. Or in this case all the supernaturals.
In a vampire game, it could be reasonably certain that, even in the territory of another clan or covenant, that the vampire with high status in the city would be regarded seriously. But in a mixed game, that high status vamp would be ignored by the Changelings and the Werewolves. Which is when you need your spokesman for the group.