As asgard1123 said beforehand, you will have to find a way to connect this hard drive to another system. You can do this through a couple of ways: (Not sure if your HDD is SATA or IDE, but all modern HDDs are SATA.)
-Plugging your drive into a SATA/IDE cable in another system just like yours was previously
-buying a SATA/IDE to usb dock which you can easily plug into another system
Also as Asgard1123 also said, you may need to edit the permissions of the drive because it may be set to be only editable by your user which no longer exists. There would be many tutorials of how to do this online depending on your operating system, etc.