What is the formal way to define "class" in ZFC? Unlike axiomatic systems deal with classes such as NBG, the term "class" is not a word in ZFC. How do I formally treat classes?
Here is an example of what I'm exactly talking about.

Example
Consider a polynomial ring $R[X]$ over a ring $R$.
We call $X$ as the "indeterminate" and write elements of $R[X]$ as $a_0+a_1X+\cdots a_n X^n$.
This is just a convention. This notation DOES NOT mean that it is a sum of $a_kX^k$, since one has NOT defined $X$.
However, $X$ can be formally defined if one defines $R[X]$ as the set of functions $\omega\rightarrow R$ with a finite support and define $X$ as $f(i)=\delta_{i1}$.

Just like this example, I want to know how to formally treat "class".
Here is a phrase in Jech-Set theory

p. 3
Although we work in ZFC which, unlike alternative axiomatic set theories, has one type of object, namely sets, we introduce the informal notion of a class.
If $\phi(x,p_1,\cdots,p_n)$ is a formula, we call $\mathbb{C}=\{x:\phi(x,p_1,\cdots,p_n)\}$ a class.

To be very precise, $\mathbb{C}$ defined here is just a drawing, not a mathematical object.
Here is an example to make clear what I'm trying to ask:

Let's assume there is a guy who only knows English.
To him, only meaningful combinations of alphabets are the meaningful words. The combination such as "adssyfeq" is not a word to him. (Hence, we can consider "he is a mathematician using sets under ZFC")
One day, he faces a combination of Chinese alphabets. Well, to him, this combination of letters is just a drawing and there is no way to make this word meaningful unless he learns Chinese which is a completely different from English.

Just like this example, is it legit to use metalanguage such as $\{x:x=x\}$ in ZFC? Is $\{x:x=x\}$ a object we can talk about in ZFC?
Also, I heard that one can treat and define classes formally if one allows inaccessible cardinal axiom. How?
Thank you in advance..
 A: You can formally define classes in the meta-theory.
If the meta-theory is a set theory, then you can think about the universe $V$ of sets as just a set model in the meta-theory's universe. In that case a class is just a definable (with parameters) subset of $V$.
If the meta-theory is some arithmetic theory, or otherwise incapable of expressing semantics as sets (meaning that we work syntactically with $\sf ZFC$ and proofs, rather than taking a model of set theory and so on), the a class is really just a formula which has possible parameters, and we say things like "The relativization of the axioms of $\sf ZFC$ to the class defined by this formula is provable from the axioms of $\sf ZFC$" (with the additional quantifiers about parameters when needed, e.g. there is a choice of parameters such that ... or for every choice of parameters ...)

Assuming an inaccessible cardinal makes things easy because it gives us a set universe which agrees with the meta-theory's universe about the power set operation, something which is not to be underestimated. But of course we can do with much much much less than that. Any set model of $\sf ZFC$ will do, it doesn't even have to be a well-founded one.
