To clarify, do you have access to the rekordbox folder (database) from the old computer? If so, browse to:
Win 7/8/10: C:\Users\yourusername\AppData\Roaming\Pioneer\rekordbox
OSX: /Users/yourusername/Library/Pioneer/rekordbox
Adjusting for the location of your OLD drive, then copy that entire rekordbox folder into the location exactly as above. Next time you open rekordbox, provided the user name is the same as it was previously, no other changes are needed and you could simply open the database (as the music is external and in the same location).
If the user name is different, a change will be needed - if that's the case, let me know and I can provide instructions.