For a collection $C$ define an object $b \in C$ iff $b$ is a class and $b \notin b$ and otherwise $b \in C$. Then we can decide any object whether it belongs to $C$. In Hungerford's Algebra p.2 it states in illustrating Gödel-Bernays axiomatic set theory that
Intuitively we consider a class to be a collection $A$ of objects such that given any object $x$ it is possible to determine whether or not $x$ is a member (or element) of $A$
Since it's possible to determine whether an object belongs to $C$, $C$ must be a class.
Then there is a paradox whether $C \in C$. How to avoid this in Gödel-Bernays axiomatic set theory.
In axiomatic class theory like Von Neumann–Gödel–Bernays, $\mathsf{NGB}$, only sets can be elements, so no proper class belongs to anything, much less itself. In a typical set theory, classes do not exist as objects that can be collected together but are instead meta-theoretic. Class theories kind of continue this perspective by limiting what we can consider as collections beyond sets. So $C$, as you defined it, wouldn't exist.
The issue with allowing us to consider any collection to exist is precisely your worry: Russell's paradox. So set and class theories like $\mathsf{ZFC}$ or $\mathsf{NGB}$ give a kind of iterative concept of collection. If we consider the collection of all sets, we don't get a set, but instead a class. And we can continue this iterative conception: if we consider the collection of all classes, we don't get a class but a 2-class. The collection of all 2-classes would be a 3-class, and so on.