How does ZFC account for uncountable subsets of naturals? I understand that the set of naturals, N, exists in ZFC. And from the power set axiom, the power set of N exists. And there are uncountably many members of P(N). 
But there are only countably many finite expressions of our set theoretic vocabulary. Does that mean there are members of P(N) that can't be expressed in the language of ZFC?
Specifically, if we take a random, infinite subset of N, how do we express the identity of this set in ZFC?
 A: Yes, there are more sets (and even more subsets of $\Bbb N$) than can be uniquely described in our language.
Note that ZFC is a theory of sets, not a theory of sets that we can write down and describe explicitly.
If $A$ is a "random, infinite subset of $\Bbb N$", then there is little we know about it (for example, is $1\in A$?), but we can still be confident that, for any set $B$ with $B\ne A$, there exists an object $x$ with $(x\in A\land x\notin B)\lor(x\notin A\land x\in B)$ to distinguish these two.
A: I think you mean to ask:

How does ZFC account for the uncountability of the set of all subsets of the naturals?

The answer is: very, very cleverly! In particular, rather than try to name every possible subset of the naturals, ZFC simply tells us how to reason about these subsets. For example, for each subset $A$ of the naturals, I can form the powerset $\mathcal{P}(A)$. This makes sense even if I can't name every $A$ to which this implies! Honestly, its mind-boggling how clever the whole notion of basing math on "axioms" really is.
But it gets weirder. It turns out that if ZFC has a model at all, then it has a countable model $\mathbf{M}$. So the element of $\mathbf{M}$ that plays the role (in $\mathbf{M}$) of the powerset of the naturals actually only has countably many elements! This is Skolem's paradox. To resolve the paradox, note that the phrase "$X$ is countable" means: "There exists an injection $X \rightarrow \mathbb{N}$." It may be the case that we have an injection $f:X \rightarrow \mathbb{N}$ in the ambient universe, but $f \notin \mathbf{M}$ despite that $X \in \mathbf{M}$. In this case, $X$ may appear uncountable from the perspective of $\mathbf{M}$, despite that you and I know that in reality, it is really countable.
I guess this weirdness is essentially the price of being clever and trying to axiomatize things. But if so, it is a worthy price, since the clever idea of basing math on axioms (rather than definitions) is what makes a foundations of mathematics possible, where otherwise, it simply wouldn't be.
A: A particularly strong instance of this is Skolem's paradox: even though ZFC proves that $\mathbb{R}$ is uncountable, there are models of ZFC containing only countably many real numbers (in fact, there are models of ZFC which are countable)!
A: Expressions in set theory can be defined as sets within set theory (inductively so), but the class-sized relation "$a$ satisfies $\varphi[x]$" can't, so you can only say "there are undefinable sets" in the metalanguage.
However, if you take a model $\mathcal{M} = (M,E)$ of ZFC, then the (set-sized) relation $\{(a,\varphi[x]) \ | \ a \in M$ and $\mathcal{M} \vDash \varphi[a]\}$ is definable, hopefully. The set of integers of $\mathcal{M}$, that is $N:= \{a \in M \ | \ \mathcal{M} \vDash a$ is a finite ordinal $\}$, need not be countable, nor does $\mathcal{P}(N):=\{a \in M \ | \ \mathcal{M} \vDash \forall x(x \in a \rightarrow x \in N)\}$ need be uncountable. For instance, both can be countable, so your argument can't work in general.
For some models we have $|\mathcal{P}(N)| > |\mathbb{N}|$ so there are elements of $M$ that $\mathcal{M}$ doesn't define.
What is important is that whereas you can generally talk about "definable sets" by identifying them with their definition and then keeping in mind that different models of ZFC may have structurally different interpretations of the set that matches the definition, a set itself ("a point of the universe") doesn't have an identity - existence without essence. I personnaly merely regard the use of undefinable sets as a syntactic step encouraged by the rule in natural deduction: if $T \vdash \varphi[v]$ for some symbol $v$, then $T \vdash \forall x \varphi[x]$.
