Do I have to purchase Adobe Premiere Pro on the laptop/computer I intend on using it?

To be more specific, you may have Premiere Pro CC installed on any number of computers, but no more than two of those may be activated at the same time on the same Creative Cloud membership. If you try to run a Creative Cloud app on a third computer, the others must be logged off before you do that (this step may be performed at the prompt when launching when too many computers are logged on to the same CC membership account).

Cloud License allows 2 activations at a time http://www.adobe.com/legal/licenses-terms.html
-Install on a 2nd computer https://community.adobe.com/t5/Get-Started/Installing-creative-cloud-on-two-devices/td-p/6048133