Is there an axiomatisation of some kind of category theory and a definition of sets in this framework such that the axioms of ZF resp. ZFC are theorems?
Yes, this is a well-known fact that goes back to Cole, Mitchell, and Osius in the 70's. The relevant kind of category is, as Harry says in the comments, a well-pointed topos with extra properties; Lawvere's Elementary Theory of the Category of Sets is an axiomatization of such a category. To define ZF-sets in such a category you build them along with their hereditary membership structure as well-founded graphs; there is a good introduction in Chapter VI of Mac Lane & Moerdijk's book Sheaves in Geometry and Logic. If you want a very detailed treatment that includes analogous results for a wide variety of set theories including both ZF and ZFC and also weaker and constructive versions, there is my own paper Comparing material and structural set theories.