The organizer stores its data in catalogs.
Physically, catalogs are folders (one folder per catalog) containing the main component, a sqlite database named 'catalog.pse8db' as well as several other components such as the thumbnail cache.
By default the catalogs are stored in 'hidden folders' in C:\ProgramData\Adobe\Elements Organizer\catalogs.
When you are in the organizer, you find the exact location of the current catalog in the Help menu, System Info.
So, do a search for 'catalog.pse8db' on the old drive.
If you can find it (make sure by its size that it's not a new empty catalog), you can copy the whole folder either in that same location in the new computer or anywhere else.
Double clicking on that catalog.pse8db file should open the organizer with this catalog.
Your problem will be that the organizer will recognize that the drive identification has been changed and show all thumbnails as 'missing'. For the organizer, the drive is defined not only by its letter, but also by the internal serial number of the drive.
So, you'll have to use the 'reconnect' function of the organizer so that the organizer finds the real location and updates the database.
http://www.johnrellis.com/psedbtool/photoshop-elements-faq.htm#_Quickly_reconnecting_large