A B767-300's APU apparently malfunctioned causing cabin smoke and then it shut down after gate arrival. The smoke in the cabin set off the LAV fire alarms.

Narrative

While approaching Taxiway Bravo we started the APU. Arriving at gate we parked the brakes and put both fuel control switches to cutoff. The APU took over the electrical power. At about half way through the checklist we smelled something burning and at about the same time there was a commotion in the back. I turned and saw at least one red light on for a lavatory fire in the forward left lavatory. Then the APU failed. No electrical power had been plugged in so we had a dark airplane with people trying to deplane. I placed the emergency power switch to ON to give them some light in the back. I heard a Flight Attendant say it seemed to be dissipating so it seemed the best course of action was to continue to deplane out the front exit. I made a PA to the passengers telling them to try to exit as quickly as possible and that we were trying to figure out what had caused the smoke. We decided to pull the APU fire handle but not rotate it since it seemed like that was what caused the problem. I then called ramp to have them install electric power to the plane; then called Maintenance to let them know of the problem. The First Officer and Relief Pilot went outside to have a look and all appeared to be normal from the outside. Later I was told all lavatory smoke lights came on in the back at the same time. [Passengers] all deplaned out forward exit. Maintenance arrived and I told them the story.