I am starting a Horror Campaign and am looking for ideas on how to start the campaign (how do the characters meet?). The tavern idea doesn't really work. I thought about having all the characters being chained to a dungeon wall, or all tied together. Give me your thoughts and a few ideas. 

Village get's eaten - only a few lucky survivors manage to get into ... like a storm cellar or something. Or they're on a stage coach going somewhere, and when they get there, everyone is either gone, or killed, or whatever. 

Or ... they're all killed, but come back (to life?) - for reasons known only to the GM, and the monster. 

Typically horror starts out normal, often in an isolated area to prevent people from running away easily.  Weird things start to happen, people investigate and things go bad FAST.

Then people begin to run and it is much much harder than it should be.   Villains almost seem to teleport ahead of them. 

What is the horror story?

Starting at a Tavern can very easily work.  Have you even seen "From Dusk Till Dawn"? Or how is starting at the Tavern any different than a group of strangers meeting up at a Summer Camp?  Now this might not be the themes you are going for, which is fine, but the point is it is hard to give suggestions without an understanding of direction the horror component of the story is heading. 

Other ideas:

1) Members of the party have received an invitation to meet a location they are unfamiliar with.  This location can be the source of the horror or they meet a creepy individual who persuades them to take part in an adventure (without revealing the depths of dangers they will encounter). 
2) The characters are just in a town going about daily business.  From there insert an event like a monstrosity appears, people start turning mad and violently attack, or they come across a rambling character how dies and sparks their interests (perhaps handing on of them a clue). 
3) They awaken in a laboratory; each strapped to a mechanism designed for experimentations that will be performed on them. 
4) Invading creatures have cause the characters to hide in a barn or basement where they realize their only hope for survival is to ban together. 
5) A shriek awakens the characters late at night.  They rush out of their separate rooms to find a resident in the inn or patron at the tavern below has died under mysterious and violent circumstances.
6) They start off on a standard quest with a group of NPCs.  When they complete the milestones to level up to 2 or 3 then introduce horror element: they release of an evil force, a discovery of creatures that force them into the Underdark, or a curse that threatens the very existence of the world. 
7) Give each character a disease or cursed effect (minor).  They individually have traveled to a Mage who instructs them they must go through this haunted land to achieve an objective that will end their alignments.  
8) Passengers on a ship or travels in a caravan.  A storm hits and they find themselves stranded in an unfamiliar land. 
9) Read something like "You lie down and close your eyes for a peaceful slumber.  Suddenly, you are awaken and find yourself in the presence of Asmodeus.  Looking around the chamber floor, you see a several strangers all in their bed clothing like yourself.  'My deepest apologies', bellows the Dark Lord, 'but there are more pressing needs than your precious night's sleep." 
10) Have the players all know each other and they have returned home for the funeral of a mutual friend. The manner in how their friend died and information some characters may have received in private correspondences result in the party having suspicions about the explanation the local magistrate has given for their friend's death. 

Again, without knowing more about the horror aspect of the game you want to run it is hard to determine if my examples can be helpful.  I think you need to start with the middle of the story first, and then work back to get to a logical starting point. 

Started my campaign in a horror setting (by the way, Van Richten's Guide is AMAZING if you are thinking about a horror campaign [and is also just really great regardless]). 

The way I started it was there was a horrible thunderstorm that travelers needed to find shelter from. Basically one of the player characters or NPCs finds a rocky outcropping or something and starts a fire, which attracts the other players. As they are huddled for warmth the weird thing happens: the rain stops and a mist the befuddles the senses rises from the ground, obscuring everything as ghostly howls and wails start to move closer. Maybe instead of a rocky outcropping they find a nearby abandoned cabin and trigger the local spirits to take umbrage with them. Maybe the cabin is the lair of a hag and their coven. Etc. 

I second the advice to read Van Richten's. There are a lot of different horror settings and tropes, and without knowing the details of your game we can't really give you what you're looking for.