Is there a way for me to auto start ccleaner and automatically Run Cleaner after I close Google chrome every time?
I can't help you on how can you accomplish the Automatically running of CCleaner after some app, like Chrome closes, but I can help you in making that easier.
As, Zangetsu said, even if you could make such an app, "it will only open CCleaner & will not automate its inside process such as "RegCleaning" etc"
So, for that I would like to add, that CCleaner supports Command Line Parameters.
So, if you edit a shortcut of CCleaner by following command, just double clicking it, would automatically start the software, and would perform the cleaning.
Source:
CCleaner - Command-line parameters
CCleaner runs silently and automatically, using the current set of saved options to clean the PC. CCleaner then exits.