Is it possible to back-up a PS3 on an Internal HardDrive?

You'll need to put your old PC internal drive in a usb drive enclosure then possibly re-partition and format it as FAT32. You can then plug it into the PS3 and backup from the internal drive to your new external drive. Since the backup/restore function is part of the PS3 software you have to boot it from the internal drive first.

Assuming your old PC drive has enough space to backup the PS3 in the first place of course.