Shortly after takeoff, and after the landing gear was raised, the pilot noticed an over-temperature of the right engine. A power reduction was made and afterwards symptoms of a compressor stall occurred. The pilot then shut down the right engine. A 180 degree turn was made. During the turn the cockpit filled with smoke with flames visible coming from the right engine area. The right fire bottle was discharged but the fire continued. When the landing gear was lowered the fire extinguished itself. It was found that the snap ring to the right engine oil temperature relief valve was missing. The right engine oil temperature relief valve was found on the floor of the engine nacelle. The right engine had been replaced 1.8 hours prior to this flight due to FOD. Engine oil had escaped after the relief valve separated. Oil had accumulated in the wheel well, wing and engine nacelle. The oil was ignited, feeding the inflight fire.