This particular version of Batman is based primarily off of the The Batman cartoon and the Beauty and the Beast fairytale. (Borrowing from the Disney version yes, but also from various other retellings.) Bruce was cursed into his monstrous form at about age twelve, and is unaware that the curse is breakable. The curse has also affected the Manor, Alfred Pennyworth, and Lucius Fox, and later affected Barbara Gordon and Dick Grayson when they tried meddling with the fabric of the spell.
Bruce is turned into a giant bat monster (a more detailed description can be found below) which comes with benefits, such as enhanced night vision, hearing, increased toughness and strength, and the ability to fly.
Alfred and Lucius were not so fortunate- Alfred has essentially been turned into a ghost, and Lucius is the batcomputer itself. Barbara and Dick have been turned into a smaller bat monster and bird monster, respectively, and are accordingly referred to as Batgirl and Robin.
The curse has a "taboo" built into it- a kind of magic that prevents someone from discussing certain topics- which restricts Bruce and the others from telling anyone their true identities or that they're under a curse.
Bruce generally goes by Batman and as far as most people are aware, Bruce Wayne died sixteen years ago. Magic keeps the Manor hidden from most outsiders, and as for the rest of Bruce's inheritance, it's either gone towards various charities or wound up in the hands of trusted associates of the Wayne family. The Batman and his family mainly live off of things they've foraged from the city, supplemented by what they can grow on the Manor grounds.
Similar to canon, this Batman and his family spend most of their time fighting wrongdoers throughout Gotham City. It's just there's usually a bit more magic involved.
The Batfamily are not on good terms with the police, especially because the Commissioner wrongly believes the Batman may have murdered his daughter. Many of the Batman's rogues still exist in some capacity or other.
Details of the Batman's relationships and situations may be changed depending on who I'm roleplaying with.
The Batman stands at roughly seven feet tall and has a muscular, humanoid figure. His body is almost entirely covered with dark fur, which is short and velvety in most areas but thickest on his chest, neck, and scalp, forming a sort of mane. On the outside of his arms are several sharp spikes, five on each arm, that are dark in coloration and can cut you very badly if you aren't careful handling them. There are similar spikes on his legs, one on each calf and one on the back of each heel.
He has the usual four fingers and a thumb that a human would have, though all end in short talons. However, each foot has three large, flexible toes that also end in talons.
Sprouting from his back are a pair of large bat wings, which he can use to fly with even though that probably shouldn't be physically possible.
His face has a bit of a snout with a prominent nose, and his teeth are numerous, small, and sharp. He has a pair of large bat ears and excellent hearing. Batman's eyes are blue, and appear unnervingly human in contrast to the rest of his appearance.
Some sketches, to help with mental references:     
These also might help, especially if you want a reference with color:   
In terms of clothes- he usually wears pants, although they're probably ripped up past the knees, thanks to the spikes on his calves. If he's out patrolling then he has a belt with various pouches on it, which contain useful odds and ends, including some magitek devices. It's essentially like his canon utility belt, although not all the equipment is the same. (He doesn't use grappling hooks as much, for instance, since he can just fly if he needs to go roof hopping.)
Other than that, he generally doesn't wear any clothing unless it's cold outside.
- tl;dr: Big buff bat furry.
Pretty close to his canon personality: The Batman devotes himself to the protection of innocents, but can be rather aloof and intimidating. He's actually pretty intelligent, but he's quiet and prone to acting almost like an actual animal (baring his teeth when he's angry, investigating unfamiliar things by sniffing them, etc.) sometimes. He'll definitely play up the whole "beast" thing if he thinks it'll frighten people, especially people he regards as threats. And having spent the last decade and a half trapped as a beast, he has come to somewhat regard himself as one. As you may have guessed there are a lot of unresolved guilt and self-esteem issues under the surface, though paradoxically Batman is also very prideful and can be self-righteous.
He does have a warmer, more playful side however. It mostly only comes out around people he's truly comfortable with (i.e. the Batfamily), and those people can expect friendly teasing and protectiveness, but he's also been known to be strangely gentle in his treatment towards crime victims and other vulnerable people he comes across. The Batman is distant, but he's not cruel- unless you manage to trigger his rare vindictive streak but again, it's reasonably rare.