As you've never seen the original website, you can't reference that. You don't know how accurate the cache is. For instance, I've seen Google Cache rearrange the formatting and strip out images from websites.
Based on that, the only place you could sensible reference is the cache file.
But, the cache file is likely to disappear very soon. It could only be live for a matter of days. I would save a copy as quickly as possible and grab images. Depending on the nature of how you want to refer to this, there can even be cases where it's appropriate to include a screenshot.
If readers will need to refer back to the information on the original site, I'd suggest archiving this somewhere online yourself (it could be on a blog). In that case, on your archive, provide the background to how this was obtained. In the paper, reference your archive.
You will also need to note somewhere in the paper that the original site was closed (and ideally a date).
I've come across this problem several times in my contract cheating research, where the sites we collected data from were closed, went offline, merged with another site, completely changed their format, or reset all their data. All it was really possible to do was to write a commentary around the changes.
You might also want to do a search for the quote you want, just to see if there's an alternative source you can reference. In some cases, it might really just be easier to not mention the website at all, unless this is absolutely crucial to the success of your paper.