Unfortunately, nonstandard models will survive any such attempt. This is guaranteed by the Löwenheim-Skolem Theorem which says that if a countable first-order theory T has an infinite model then it has one of every infinite cardinality. Since an uncountable model necessarily has nonstandard elements, this guarantees that there is a nonstandard model of T (and even countable ones).
Actually, in your case you need a "two-cardinal" version of Löwenheim-Skolem. In your ZFC example, you move to a theory which interprets arithmetic inside a definable substructure (the set ω). The definable substructure of such a model which might still be countable even if the model itself is uncountable. Nevertheless, one can still blow up the size of the natural number substructure via the ultrapower construction, for example.
To evade the Löwenheim-Skolem Theorem, one has to move beyond first-order logic. For example, in infinitary logic one allows infinite disjunctions such as
$$\forall x(x = 0 \lor x = S0 \lor x = SS0 \lor \cdots)$$
which ensures that the model is standard. Also, second-order allows quantification over arbitrary sets under the standard interpretation, which again prohibits non-standard models. (See this related question.) This is the characterization of N most commonly used by working mathematicians.
There is a cheap way of doing this, which may not be the optimal approach when a subtle task (such as the foundational question you have in mind) is the goal. But, then again, this may suffice.
Working in an appropriately strong theory, to simplify, the standard way to check that NBG is conservative over ZFC is to see that any model $M$ of ZFC can be extended to a model $N$ of NBG in such a way that the "sets" of $N$ give us back $M$. Again to simplify, assume the model $M$ is transitive. The model $N$ we associate to it is Gödel's $\mathop{\rm Def}(M)$, the collection of subsets of $M$ that are first order definable in $M$ from parameters (The proper classes are the elements of $\mathop{\rm Def}(M)\setminus M$.)
This suggests the simple solution of defining the models of "iterated-NBG" as the result of iterating Gödel's operation. So, given a transitive model $M$ of ZFC, the $\alpha$-th iterate would simply be what we usually denote $L_\alpha(M)$.
I am restricting to transitive models here, but there is a natural first order theory associated to each stage of the iteration just described (at least, for "many" $\alpha$), and I guess one could try to axiomatize it decently if enough pressure is applied.
There are some subtleties in play here. One is that most likely we want to stop the iteration way before we run into serious technicalities ($\alpha$ would have to be a recursive ordinal, for one thing, but I suspect we wouldn't want to venture much beyond the $\omega$-th iteration). Another is that the objects we obtain with this procedure would have wildly varying properties depending on specific properties of $M$.
For example, if $M$ is the least transitive model of set theory, then we "quickly" add a bijection between $M$ and $\omega$. In general, if $M$ is least with some (first order in the set theoretic universe) property, then we "quickly" add a bijection between $M$ and the size of the parameters required to describe this property (this is an old fine-structural observation. "Quickly" can be made pedantically precise, but let me leave it as is).
So you may want to work not with ZFC proper but with a slightly stronger theory (something like ZFC + "there is a transitive model of ZFC" + "there is a transitive model of "ZFC+there is a transitive model of ZFC"" + ...) if you want some stability on the theory of the transitive models produced this way. (Of course, this is an issue of specific models, not of the "iterated-NBG" theory per se).
I should add that I do not know of any serious work in the setting I've suggested, with two exceptions. One, in his book on Class Forcing, Sy Friedman briefly mentions a version of "Hyperclass forcing" appropriate to solve some questions that appear in a natural fashion once we show, for example, that no class forcing over $L$ can add $0^\sharp$. The second is by Reinhardt in the context of large cardinals and elementary embeddings, and is described by Maddy in her article "Believing the axioms. II". As far as I remember, neither work goes beyond hyperclasses, i.e., classes of classes.
Best Answer
According to Godel's incompleteness theorem, ZFC cannot prove its own consistency. Therefore, it is relatively consistent with ZFC that there are not any set models of ZFC. In this case, there is still a proper class model of ZFC, namely the von Neumann universe, V, itself, among others (i.e. L, forcing extensions of V). However, the fact that V is a model of ZFC cannot be proven formally within ZFC. Indeed, truth in V cannot be defined in V due to a result of Tarski.
If we allow for some stronger axioms, then we can get set models of ZFC. For instance, if there exists an inaccessible cardinal, $\kappa$, then $V_\kappa$ is a set model of ZFC.