If the new computer is trying to boot from the old hard drive, try to switch the HDD's SATA ports. Since it probably used to be your system drive, it contains a boot partition and if it is plugged into a SATA port with "higher boot priority" than your new drive, the system will try to boot from it. Boot priority can vary from board to board; some will try to boot from PATA first, then SATA and prioritize SATA drive with lower ID (ex: SATA1 before SATA3). If your MB has multiple SATA controllers, it might also prioritize the primary one before the secondary (usually, the SATA ports will have color coding to differenciate).
If all that fails, you can always try to use a Linux LiveCD to boot and copy files from one drive to the other and then simply remove the old drive.
If that doesn't work either, you can also try a USB HDD adapter like these
. Just remove the old HDD, and plug the HDD using the USB adapter, I think you would see the HDD as a USB removable drive once Windows booted.