Let's keep adding once undecidable statements

Short answer: $ZFC_\alpha$ only makes sense for those $\alpha$ which have "nicely definable representations" - specifically, for the computable ordinals $\alpha$. (An ordinal is computable if there is a binary relation on $\omega$ which is computable, and which well-orders $\omega$ with order-type $\alpha$.) Every computable ordinal is countable, as is the least noncomputable ordinal $\omega_1^{CK}$ - so this process stops long before reaching $\omega_1$, let alone going through all of $ON$.

The details of this process are quite subtle - Sacks' book "Higher Recursion Theory" has a very good treatment in the first chapter. Especially relevant is the representation of computable ordinals via ordinal notations, and Kleene's $\mathcal{O}$. Basically, Kleene's $\mathcal{O}$ is (in a precise sense) the set of all names for computable ordinals. There's a lot of incomputability here: $\mathcal{O}$ is incomputable (in fact $\Pi^1_1$ complete!), as are the relations "codes the same ordinal as," "codes a bigger ordinal than," etc. Moreover, the natural structure of $\mathcal{O}$ is as a partial order, not a linear order, reflecting these undecidabilities. The object $ZFC_{\omega_1^{CK}}$ should, intuitively, be the result of iterating the consistency process along a "maximal" path through $\mathcal{O}$; however, there are many different maximal paths, including "short" ones (i.e. ones of order-type $\omega^2$ instead of $\omega_1^{CK}$). The study of paths through $\mathcal{O}$ is an extremely rich area of computability theory and descriptive set theory.

Tl;dr: it's a very good idea, but it's a bit more complicated than that.

To add to the references which have already been given, I'd like to point out that Torkel Franzén wrote an entire book (at a fairly elementary level, and with a number of IMHO rather interesting philosophical digressions) pretty much about answering this exact question: Inexhaustibility, A Non-Exhaustive Treatment, ASL Lecture Notes in Logic 16 (2004). Especially see chapters 13 ("Iterated Consistency"), 14 ("Iterated Reflection") and 15 ("Iterated iteration and inexhaustibility"): they include accounts of the results by Turing, Feferman and Franzén mentioned in Payam Seraji's answer.

There's some nice exposition of this topic here:

https://xorshammer.com/2009/03/23/what-happens-when-you-iterate-godels-theorem/