While a wide variety of creatures work as RPG protagonists in the generic sense, horror-themed RPGs tend to use inhuman protagonists as a way to internalize the fundamental thematic elements.
Vampire's Formula: "Personal Horror"
White Wolf's design for Vampire: the Masquerade was based on an idea they called "personal horror": you are a vampire, and always will be. Like body horror, personal horror gets a lot of its power from the feeling of corruption and inevitability, though instead of visceral disgust and violation it's generally playing towards the emotional and tragic.
Sure, your vampire PC might encounter and defeat all kinds of awful monstrous things, but one thing you can't escape is that you yourself are corrupted and monstrous. Losing control to the Beast, the slippery-slope morality mechanics, the rules for having drink blood to use your powers — all of these are around to remind the players that just living as a vampire involves hurting people all the time; however moral and altruistic you may try to be, the bedrock of your life is exploiting and abusing others.
The Monsters You Play Are Also Victims
Vampires are also compelling because they used to be human. A Vampire PC's "unlife" started with someone else turning them into the thing they are now. And from your sire you inherit not only the power and curse of vampirism but also membership in a culture of clans and elders, a society that exists mainly to perpetuate itself. Vampire society is there to continuously tell you that you ought to be dehumanizing everyone below you (they're "kine"), that trying to live your life any other way is doomed and fruitless. (At its core, I think this setup was a pretty smart take on the moral unease experienced by many middle-class teens: Am I just going to end up a soulless parasite? Why did everyone before me just decide to fall in line and turn a blind eye to the suffering of others?)
Monstrous nature represents indelible psychological scars in many other games:
NWoD Changeling has been described as a game that's really about abuse victims trying to find their way back into normal lives.
Prometheans are, by their nature, forlorn and isolated, struggling for human connection.
Wraith's themes are most clearly expressed in Charnel Houses of Europe: How can you move past a hellish experience? Will it define you forever in ways you can't control?
Powerful and Powerless, Victim and Victimizer
I think a good template for monstrous PCs starts with these elements:
The character's monstrous nature is a source of strength. Even if you're not playing a game about supernatural badasses, it's important for the inhuman part of the PC to be useful in accomplishing goals, because this creates a reason for players to continually bring it up in play.
A lot of the core horror in horror fiction is based on powerlessness and suffering. The monstrous protagonist is still a protagonist. Part of the price of admission to all that monstrous power is that it's also your weakness and your burden.
Horror monsters defile and abuse. That doesn't change just because we decided to make them the protagonists. If the players want to be horrible monsters, make them do horrible things. It's their nature. It sustains their existence. It makes them exciting and tragic.
Monsterhearts interprets supernatural teen drama (like Vampire Diaries) through a lens like this, representing different forms of emotional imbalance as vampires, werewolves, witches, &c. Their nature both empowers the characters and hurts them, and can easily lead them to hurt other people.
Note that the horror monsters needn't be a full-fledged species with their own civilization and such. You can just easily play a game where the monsters are rarer and never spend their time relating to any NPCs of "their own kind" (imagine The Crow if it was about three friends who were all brutally murdered by the same people, for instance). The important thing is that they have lots of ways to relate to people.