I did see a tutorial on recording from your 360 and the Easycap was used but not in the same configuration you're talking of. They had the 360 - TV via SCART/RCA then from the TV's RCA Out, to the Easycap - to PC/Laptop.
So if I understand it right, you're monitor will need to output to the Easycap using the Composite video RCA and then using your splitter lead, the audio will go direct from your XBox to the Easycap. Not sure if it will work but there's a dedicated forum for Easycap here
http://forum.easycap.co.uk/