I was just wondering if this erased old files that were previously deleted.
I am selling my computer and don't want anyone to be able to access those
old files. I know that it can be done. If ccleaners doesn't, what can I do to
clean them?

You need something completely different to wipe all that data such as DBAN. Note you'll have to still have the Windows install CD handy to reinstall the operating system.

Edit:
If you no longer have the Windows install CD, about the only other option would be to use Eraser.