This answer will be written assuming that you are coming from an amateur or undergraduate understanding of axioms, as is used in much of mathematics outside the study of mathematical logic. I will try not to say anything false, but I can't tell "the whole story" at this level. Please let me know in comments about areas of the answer that remain confusing.
I will call this common, informal reasoning "ordinary mathematics". It includes many informal principles for reasoning about mathematical objects and sets of mathematical objects. (In fact, many published mathematics papers outside mathematical logic work entirely in this informal system, so it is not just for amateurs and undergraduates.)
Characterizing the reals
In ordinary mathematics, there is a set of axioms that characterizes the rationals: they are the so-called field of fractions of the natural numbers. Of course, this doesn't "completely" characterize the rationals, because we still have to characterize the natural numbers. But I think this is good enough for the question at hand.
Now, given the rational numbers, there are ways to construct the real line axiomatically. One way is via Dedekind cuts, and in this context we can easily verify that the collection of Dedekind cuts of rationals satisfies the axiom of completeness and has the other properties we expect for the real line.
If we continue working in the same framework of ordinary mathematics, no additional axioms are needed to show that the reals are uncountable. We can prove many things about the reals using ordinary mathematical methods and the fact that the reals satisfy the axiom of completeness. For example, we can prove that there is no real number that is greater than every natural number (see below). We abbreviate that last fact by saying that the reals are an Archimedean field.
Many real analysis textbooks use only ordinary mathematical reasoning, along with the fact that the real numbers are a complete Archimedean ordered field, to prove all their results. One of these results is that the real numbers are uncountable. This can be proved in many ways, and I recommend picking up a good introductory textbook on real analysis to learn the details. I have used Understanding Analysis by Stephen Abbott, which I think is understandable and has a nice introduction to many properties of the reals.
In mathematical logic, one thing that we do it to try to formalize the principles of "ordinary mathematics", to see what formal axioms are actually needed. The seeds of this project were planted as early as Euclid, but it fully blossomed from the late 1800s through the early 1900s. Indeed, most of the major issues were understood by 1940.
What we learned is that no "formal system of axioms", in the sense that you would naively expect, can fully characterize the natural numbers, the rationals, the reals, or any other particular infinite structure. It turns out, because of the "finite" nature of a formal proof, that formal proofs are simply incapable of fully capturing the nuances of infinite structures.
We can certainly write formal axioms that are true about the natural numbers, but these axioms will also be true of other stuctures, which we call "nonstandard" because they aren't the structure we really wanted to study. So the axioms won't characterize the naturals. For purpose of ordinary mathematics, though, we can ignore the nonstandard models, and know that whatever we prove in our formal systems really is true about the "standard" models we care about.
Now, you may ask, how can we characterize the rationals in ordinary mathematics, but not in any formal system? The answer is essentially that, in ordinary mathematics, we begin with the set of natural numbers as a given object. Similarly, we take other strucutres for granted, such as the powerset of the naturals. Relative to these starting objects, we can characterize the rationals, the reals, etc. In formal reasoning, we do not have these given structures to use as a reference, and there is no way to fully capture them with formal axioms.
Here is the promised argument that the set of natural numbers is unbounded in the reals, using only the axiom of completeness and ordinary mathematical reasoning. If the naturals were a bounded set in the reals, they would have a least upper bound, which we can call $s$. Then there must be some natural number in the interval $(s-1, s)$, or else $s-1$ would be a smaller upper bound than $s$, which is impossible because $s$ is the smallest upper bound. But, if there is a natural number $n$ in the interval $(s-1,s)$, then $n+1$ is a natural number greater than $s$, which is impossible if $s$ is an upper bound for the naturals. Overall, we see that the natural numbers cannot be a bounded set.