When CCleaner installs but then won't run it's usually because it's being blocked by your antivirus.
I'd try the following in this order to see if you can find what the problem is and fix it:
Uninstall CCleaner and download/install it again from here: https://www.piriform.com/ccleaner/builds
(Use the 'Slim' installer, 3rd one down. It doesn't contain the bundled offers).
See if it will run/register now.
Check that CCleaner has installed correctly.
In File Explorer look in C:\Program Files\CCleaner to make sure that CCleaner has actually been installed.
Try double clicking on CCleaner64.exe to see if it will run from there.
Make sure that the AV is up to date.
It's best to check the AV's website, some will sometimes say they are up to date but there is a later version available on the website.
Then try running CCleaner again.
Try disabling (or even uninstalling) the AV to see if CCleaner will then run.
If it does then it indicates that it is the AV blocking CCleaner and you will need to make CCleaner an exception in the AV.
Boot the computer into Safe Mode and see if CCleaner will run in Safe Mode.
If it does that indicates that something is blocking it in 'normal' mode. (Usually the AV, but possibly some other security software).
For our further info:
What version of Windows is the computer running?
What antivirus is in use on it? (and at what version).
For your further info:
Here are the support articles for problems when installing or registering CCleaner (although they don't quite cover your situation):
Problems installing CCleaner
Porblems registering CCleaner
PS. Although not relevant to this install problem, you may have been better buying a 'Professional Plus' bundle which licences up to 3 computers rather than buying 2 individual licences (maybe not as it looks like the second one may have been the Boxing Day offer?):
https://support.piriform.com/hc/en-us/articles/360028350711-How-many-computers-can-I-install-CCleaner-Professional-Plus-on-#how-many-computers-can-i-install-ccleaner-professional-plus-on–0-0