Thomas Jech, such as many other mathematicians, demonstrates $AC \rightarrow ZL$ via transfinite induction. He says:
Proof. We construct (using a choice function for nonempty sets of P), a chain in P that leads to a maximal element of P. We let, by induction, $a_\alpha =$ an element of P such that $a_\alpha > a_\beta$ for every $\beta < \alpha$ if there is one. Clearly, if $\alpha > 0$ is a limit ordinal, then $C_\alpha = \{a_\beta : \beta < \alpha \}$ is a chain in P and $a_\alpha$ exists by the assumption. Eventually, there is $\theta$ such that there is no $a_{\theta + 1} \in P$, $a_{\theta + 1} > a_\theta$. Thus $a_{\theta}$ is a maximal element of P.
I have a question regarding this demonstration: Under what argument does Jech justify the existence of $\theta$?
In another question, Asaf Karagila argues that this process of choosing and adding elements to the chain must stop because, otherwise, we would have an injection from the proper class of Ordinals to P. Why can't this be?, why can't we have an injection from a proper class to a set?
I'm sorry if this questions seem naive, I'm still quite immature in my mathematical knowledge.
This is known as Hartogs theorem.
In particular, it follows that there is no injection from the class of ordinals into a set.
The proof is not very difficult, but not quite trivial. It has been covered over this site several times. The idea is that you consider subsets which can be well-ordered, collect those of the same order type into equivalence classes, and construct a well-ordering which cannot be injected into $X$ itself. This well-ordering is a set, and therefore isomorphic to an ordinal.
But one can make other arguments, equally compelling.
Assume that there is an injection from the class of ordinals into $X$. Consider its range $Y$, then by separation $Y$ is a set. By replacement it follows that the class of ordinals is a set, which is a contradiction.