Only a few characters are known at this point, they are...
You play as him. The player and his friends (including Hatty) were sailing on the S.S. Friendship, until a harsh storm shipwrecked them on an unknown island. Seeking shelter, the Player showed up just in time to see the Cats force Hatty to put on a glowing top hat.
He is then sent through various deadly obstacle courses throughout the game, which make up the levels of Battleblock Theater. He seems to possess a strong will, going through the increasingly-ridiculous levels without complaining once. Finally, he makes it through the Vault and rescues Hatty, before then escaping the island altogether.
The Player can be any prisoner they've rescued from the Gift Shop, though cutscenes show them as the default round head (known in the Steam inventory as Mom).
The Player's best (and most attractive) friend who was given a glowing red hat, and according to the narrator, "Ooh that's bad. Glowing red things are always bad." He seems to be the captain of the boat. Hatty later becomes corrupted by the hat and the gems he can make from the theater and becomes its owner, leading the cats. He does nothing but sit in his office among a pile of gems.
In the end, Hatty did not pay attention or maintenance to the staff of the theater, the cats, so they became very disorganized and angry. They rebelled and destroyed the theater. Hatty at this point seemed lifeless (like a doll) and did not respond to anyone around him. This is because of the conflict between the souls of the hat, and the trauma that Hatty has been through and has witnessed his friends go through, making him emotionally torn (Hence the constant stream of tears) and putting him in a catatonic state.
The player brings him aboard the boat, but is unable to make him respond. Shortly after the song "You Were My Hattington", a large wave knocks him off the boat as Buckle Your Pants starts playing. After the song is finished, he's shown sinking to the bottom of the ocean, only for the hat to land on his head again. Instead of glowing red, it glows green this time. Hatty yells "NO!" several times, before a huge laser is shot out of the hat and into the sky. Amongst other things, it hits the S.S. Friendship, a plane, and a bird before hitting a giant space beaver. The game then returns to the title screen, without explaining what happened.
They are the cats who run the theater. They wear brown uniforms that resemble real life guards. They stand taller than the player, and are relatively fat. Their exact origins are unknown, and not even the Narrator is sure where they came from, though they're shown acting as performers back in the days when Purrham Furbottom ran the theater.
In the present day, they seem to have gone insane. They rule over the bleak, decaying theater with an iron fist, forcing the prisoners to go through increasingly difficult challenges for their own monetary gain. The cats bear a huge love for Golden Yarn, and will give any prisoner who can bring them five Golden Yarns a new weapon.
When Hatty proves to be an inept manager, they start rioting and causing even worse damage to the theater, before finally attempting to kill Hatty and find someone else to take his place. At the end of the game, the prisoners manage a breakout, and several cats are shot dead as the S.S. Friendship leaves. It remains unknown if there were any survivors.
Hatty has enslaved the rest of your friends and has made you compete, fight, and you and them must work together to survive the deadly performances for amusement. They are other prisoners locked up in the Gift Shop, which you can unlock for an increasing rate starting at 10 gems. They serve as customization for the Player.
The Narrator talks to you throughout the game with many different phrases, such as "Milk, milk, lemonade, get that hat so you'll get paid" and "Don't get zapped by those things. Some people poop themselves and die. In that order!" He also narrates cutscenes, and apparently is the one controlling the cutscenes stick puppets. He also sings the "Buckle Your Pants" song and can be turned off in the settings menu.
Purrham Furbottom was a self-made millionaire, who lived in the year "17095-ish", according to the Narrator. Having amassed a fortune, he built a new attraction: Battleblock Theater. The theater stared deadly stage performances, the likes of which the world had never seen before. It was a huge success, raking in enormous amounts of money for Purrham.
However, for unknown reasons, Furbottom hated intermissions with a burning passion, and didn't allow them in the theater. He sat there for days on end, gorging himself on popcorn and never taking a potty break. This ultimately proved his downfall. Furbottom tragically pooped himself to death while running to the bathroom. According to the Narrator, he "tried to hold it in, but his little cheeks just couldn't handle it".
After his death, the cats threw his foul-smelling body into the ocean, where it was immediately devoured by sharks. He left behind nothing but his theater, his cats, and his top hat.