For the sake of clarity, the Yellow/Red/White connections are a RCA Composite cable. A Component cable is only for video and uses Red,Blue, and Green connectors. It's an important distinction as the two are not directly compatible. (Though some Flat screens accept composite inputs in their component plugs to save space on their rear panel).
DVI uses the same signal as HDMI; it's effectively just wired up different (and HDMI has audio). I can't find any adapters which accept a DVI connection, however, something like
this one accepts HDMI. You can get a DVI->HDMI adapter for relatively cheap, something like
this would be ideal in this case, (I'm sure that can be found elsewhere of course, that's just an example) I think.
Since the PC won't know the capabilities of the display, you'll want to reduce the resolution to 640x480 for the secondary display.
The trouble with a passive adapter is that the graphics card will have to explicitly support it. For the most part newer graphics cards don't sent the required analog signals that are passively converted to Composite, so an active converter is a more reliable option.