If your monitor has different inputs, then you should use that, like a VGA connector and a DVI, hook up the DVI to your PC and the VGA to the 360 using a 360-VGA cable. The connections on the front of your PC should probably be ignored, I very much doubt these are inputs, probably outputs for your TV card going to a scart type input on a TV.
Besides, the quality you'd get through phono/scart would be terrible, probably look worse than with an old TV. Using the input on my TV card, with a GameCube, the quality was abysmal.
I would suggest picking up a cheap CRT monitor and 360-VGA cable, the quality from that is exceptional, and CRT's are practically free these days. You would need external speakers though, like a cheap pair of PC speakers would do it.