Ok..first off my method is using "2" different inputs into your monitor and not "channeling" the xbox though the pc. More then likely, with your old pc, you are usingg the DVI output of your graphics card to go HDMI. Google search HDMI for more information.
I looked at your mointor again and you have 2 HDMI ports. So you can hook your PC and XBOX to hdmi.
Now the question is, do you have a 360 with HDMI or an orginal Xbox. I assumed a 360.
You will need a HDMI...