Starting from the axiom of choice and passing the lemma of Zorn we arrive at the well-ordering theorem which states that for every set $X$ there exists a well-ordering with domain $X$.
I am aware that possibly nobody will ever be able to construct an explicit well-ordering of $\mathbb R$. There are in fact enough questions to be found here on math.stackexchange.com
But what about the "next best thing": an explicit (=constructed) well-ordering of $\mathbb Q$?
As suggested in the comments, all you need is an injective function from rationals into naturals.
For example, the function $f(p/q)=2^{p/d}3^{q/d}$ where $d$ is the greatest common denominator of $p$ and $q$ is quite explicit, I would say (as explicit as one can hope for, given ambiguity of representations of rational numbers!).
Then for any two rational numbers $q_1,q_2$, declare $q_1\prec q_2\iff f(q_1)<f(q_2)$.