I'm not sure what your last two paragraphs mean, but your main question seems to be: "What axioms do you need to prove that the reals - thought of as the set of equivalence classes of Cauchy sequences of rational numbers - are uncountable?"
Well, first, note that we need some axioms to even talk about the reals defined in this manner - we need to be able to make sense of sets of sets of rationals. Different definitions of the reals may have different "axiomatic overhead." But let's leave this point alone for the moment.
Usually, Cantor's diagonal argument is presented as acting on decimal or binary expansions - this is just an instance of picking a canonical representative from each equivalence class of Cauchy sequences. If $f: \mathbb{N}\rightarrow\mathbb{R}$ is a purported bijection, we want - for each $n\in\mathbb{N}$ - to pick a representative $(a_i^n)$ of the real $f(n)$. You might worry that there's some axiom of choice shenanigans here, but that's not so - since $\mathbb{Q}$ is explicitly countable, we can do this without choice . . .
Assuming we know how fast our Cauchy sequences converge! Decimal, binary, etc. representations have a wonderful feature - they converge quickly. A modulus of convergence for a Cauchy sequence $(c_i)$ is a sequence of naturals $(k_n)$ such that - for $i, j>k_n$ - we have $d(c_i, c_j)<2^{-n}$. Unfortunately, once we start talking about moduli, things get tricky: the statement "every Cauchy sequence has a modulus" is not computably true, that is, there are computable Cauchy sequences without computable moduli. In order to make things go the way they should, we need axioms which go beyond the computable. Fortunately, they don't have to go much beyond the computable - it's enough for every function's range to exist.
So in this sense, Cantor's diagonal argument does need some nontrivial axiomatic strength. If you're interested in this sort of thing, you should look at Reverse Mathematics. However, I'll point out two things:
The axioms needed are, while going beyond computable mathematics, extremely weak.
The need to go beyond computable mathematics is mostly a function of having picked the wrong definition of real number - if we care about the computable content of our axioms, we should define real numbers as equivalence classes of Cauchy sequences equipped with moduli of continuity. Note that this is exactly what a decimal (or binary, etc.) representation is! So this is actually a very natural thing to do. In particular, if we define the reals this way, then we don't need any axioms beyond computable mathematics, and in fact even that's overkill.
Hope this helps!
