How would I go about restoring my system properly?
You boot from the cloned backup via holding down the OPTION key during startup and selecting the clone.
Once the cloned Mac OS X has fully loaded, start Disk Utility (Spotlight will find it), erase the volume/partition with the defunct Mac OS X installation.
Then open CarbonCopyCloner or SuperDuper (whichever application you took in the first place) and choose the newly formatted (erased) partition/volume as target and clone the working backup back to it.