Step 1: Make sure the cable you are using works properly and the DHCP server is good by connecting with second computer using the cable you have plugged into the problem computer. If it works, move on.
Step 2:
Right click my computer-->Manage-->Services
and Applications-->Services
Look for DHCP Client and make sure this service is running. If it is then move on, if not then turn it on and set it to automatic in the properties dialog. If it is not even there then you need to install it through Programs-->Windows Features.
Step2:
test adapter. Download a linux live CD
http://www.ubuntu.com/desktop/get-ubuntu/download and boot into linux off of the CD (no need to install anything, this is just a temporary environment running off of the CD outside of windows. A linux live CD already has a generic driver for nearly every ethernet adapter out there so just boot up and see if it connects to the internet. If you are able to do this through the linux live CD then you know it is software. Try updating driver from manufacturers website. If this doesn't work you may want to reinstall windows at this point, as any more troubleshooting would require a paid technician to help you and even they might just say reinstall.
Things to research if you want to make this a project:
Linux Live CD
WinSock Catalogs
DHCP server/client
