Step 1: Remove the hard drive from from the dead computer. (That usually involves nothing more than undoing a few screws and unplugging the connectors unless it's a ruddy Dell; they're a real pain to get into!)
Step 2: Determine whether it's an IDE drive or a SATA drive. (That usually involves nothing more than reading the label on it. If you're not sure, see here:
http://www.diffen.com/difference/IDE_vs_SATA )
Step 3: Buy a USB enclose for a 3.5" IDE or SATA drive, as applicable, from one of the usual sources. (eBay, eBuyer, Amazon, etc.)
Step 4: Connect the drive up by following the simple instructions that come with the enclosure.
Step 5. Connect the drive to your new computer using the USB connection.
Step 6: Discover that you still can't see much of the data on the drive using Windows Explorer. Don't panic!
Step 7: Take ownership of the drive, following the instructions in the video below. (It refers to Vista but the process should be reasonably similar under Windows 8 - I hope!).
Step 8: Access your files from the old drive in exactly the same way as you would from a folder on your new drive.