Suppose $C$ is a proper class and that we are working in the universe $U$ which satisfies the axioms of ZFC. Is it trivially true that $C \subseteq U$?
I think the answer to this is yes, since $U$ contains all possible sets that we could ever consider. So if a set is in $C$, then it must be in $U$.
Pretty much. If $U$ is a model of ZFC, then a proper class is just a collection of members of $U$ that does not itself form a member of $U$ (generally because it's "too big"). So yes, when we talk about a proper class, we're necessarily talking about a collection of members of our underlying model (what you're referring to as a "universe").