Not without external hardware, and even then it might not be feasible. If you want to record gameplay, an HDMI capture card for PC is probably best.
The TV encoder in the Xbox 360 can not output both composite and HDMI at the same time because it can only handle one output resolution at a time. Simultaneous HDMI and composite output would have forced it to output 480i over HDMI as well. Analog component
was also left out as a consequence, in spite of having common resolutions. So, they simply turn off the analog video DACs when HDMI is detected.