THE CREW WAS NOT AWARE AN APU FIRE HAD OCCURRED AT THE GATE PRIOR TO STARTING THE MAIN ENGINES. AFTER DEPARTURE THE AIRPLANE EXPERIENCED SEVERE VIBRATIONS; AN EMERGENCY WAS DECLARED, AND THE FLIGHT LANDED WITHOUT INCIDENT. ABOUT 1/2 OF THE UNDERSIDE OF THE ELEVATOR & TRIM TAB WERE SEVERELY FIRE DAMAGED. THE TAIL CONE WAS CHARRED & BLISTERED OVER A 3 FT BY 3 FT AREA. FOUR GROUND SERVICE PERSONNEL HANDLED THE AIRCRAFT DURING PUSHBACK. THE APU COMBUSTOR CAP DRAIN LINE WAS FOUND CLOGGED WITH SEVERAL PIECES OF ROOM TEMPERATURE VULCANIZING (RTV) MATERIAL (POTTING COMPOUND) USED IN THE APU BUILDUP PROCESS. THE APU MANUFACTURER'S OVERHAUL MANUAL CAUTIONS OPERATORS TO USE RTV SPARINGLY, AND TO REMOVE ANY EXCESS PRIOR TO INSTALLING EXHAUST PIPE SHIMS TO PREVENT PLUGGED FUEL DRAIN FITTINGS AND TORCHING STARTS. IN ADDITION, THE ATOMIZER ASSEMBLY SHOWED STREAKING IN THE PRIMARY ORIFICES, AND EXAMINATION OF THE OIL PUMP REVEALED SLOW SEQUENCING (IGNITION AND FUEL INITIATION).