No, you cannot use it as a display for your Xbox. The vast majority of laptops only have a connector for video output. Only a handful of gaming laptops supports video input (through HDMI) and those are basically Alienware laptops.
Using a capture card is possible, but I am pretty sure you are going to have lag issues.
Your best bet is to buy an inexpensive monitor for it. I assume your Xbox 360 has a HDMI port. Below is a link to some 1080p monitors with HDMI input and built-in speakers.
http
/www.newegg.com/Product/ProductList.aspx?Submit=ENE&N=100160979%20600012673%20600012696%20600029504&IsNodeId=1&bop=And&Order=PRICE&PageSize=30