Alienware X51 and Xbox 360

Hello, this is somewhat of a pc and console gaming question combined so I figured that this general discussion would be the best place for it.

To start I have the 2nd level Alienware X51 and a Xbox 360 slim. I bought the computer for gaming mostly and it is a very good machine. However, since I bought the computer I've wanted to start recording my Xbox 360 gameplay on it.

I understand the basics of computers however I have never modified or added anything to a store bought computer be sides RAM (let alone built a custom pc).

Anyway before I get too off track I better just ask my question. Since I have an hdmi port in the back of my computer tower, would it be possible to hook my Xbox 360 into the port and play/record the gameplay using my computer?

Obviously this question has been asked before and I have done a search. The only reason I feel the need to bring up the topic myself is because all of the posts I have read so far were about hooking it up to a monitor or laptop. And none of which seemed to be made for gaming like mine.

I also realize that a good way to find out would be to try it. I am just afraid if I do without asking someone more educated in electronics I might mess up one, if not both, of my systems.

Any help or feedback is appreciated!!

Thanks,
Chad