Can I transfer Photoshop CS5 from one computer to the next without the original package?

you should properly install ps cs5 on your new computer using the installation files (linked below).  you just need your serial number and the install file.

if you no longer need ps cs5 on your old computer or you've already used your two allowed activatations, deactivate (open ps>help>deactivate) it on your old computer.

Downloads available:

• Suites and Programs:  CC 2014 | CC | CS6 | CS5.5 | CS5 | CS4 | CS3
• Acrobat:  XI, X | 9,8 | 9 standard
• Premiere Elements:  13 | 12 | 11, 10 | 9, 8, 7 win | 8 mac | 7 mac
• Photoshop Elements:  13 |12 | 11, 10 | 9,8,7 win | 8 mac | 7 mac
• Lightroom:  5.7| 5 | 4 | 3
• Captivate:  8 | 7 | 6 | 5
• Contribute:  CS5 | CS4, CS3

Download and installation help for Adobe links

Download and installation help for Prodesigntools links are listed on most linked pages.  They are critical; especially steps 1, 2 and 3.  If you click a link that does not have those steps listed, open a second window using the Lightroom 3 link to see those 'Important Instructions'.