Sounds like you have a USB ADSL modem.
You can connect the Ethernet from the Xbox 360 to the Ethernet on your PC.
You will need to use Internet Connection Sharing and the PC will need to be on when you want to use Xbox Live.
More info here:
http://www.xbox.com/en-US/support/connecttolive/xbox360/connectionmethods/windowsics.htm
It would probably be better if you had a router that you could connect to as this is a dedicated device as opposed to a PC but it will work.
Routers are so cheap nowadays though, may be easier to just get a wireless router.