For instance, I've heard that he wrote an 8 volume history of Carthage prior to becoming emperor. I know none of his works are extant, but I was wondering how much about them we can piece together from quotations in other works, etc.
To my knowledge exactly zero fragments of any of Claudius' writings survive. No author quotes Claudius at all and I am unaware of any papyrus finds. The Packhard Humanities Project, which catalogs Latin authors pretty well, has no page on him and collects no fragments of his.