First let's give a better definition of this preorder: $\sf ZF\vdash\varphi\rightarrow\psi$. Then the equivalence relation is $\sf ZF\vdash\varphi\leftrightarrow\psi$.
There are infinite chains, that much is easy to prove. First note that if we allow parameters, like $\kappa$ and $\lambda$, then $\sf DC_\kappa$ implies $\sf DC_\lambda$ if $\lambda\leq\kappa$. Since we don't allow parameters here, we have to reduce to definable $\aleph$'s, like $\aleph_0,\aleph_1,\ldots,\aleph_n$ and so on. This already gives you at least one chain whose length is infinite. One interesting question is to ask what is its length, but this seems to me equivalent (or at least closely related) to asking what is the proof theoretic ordinal of $\sf ZF$, and the answer I suspect is unknown.
Finite maximal chains are tricky objects, and I suspect they do not exist. The reason is that if $\varphi$ is neither provable from $\sf ZF$ nor it proves $\sf AC$, then it is usually the case that we can extend it "a little", by adding some of the missing choice to a small family of objects. For example, choice from families of finite sets doesn't imply the axiom of choice in full. We can, however, extend it a little bit by requiring choice from countable families of infinite sets; then from families of size $\aleph_1$ and so on. Maximal finite chain would have to be some sort of particular principle that extending it even a little will necessarily prove the axiom of choice. I can't think of one.
As for antichains, those are harder to investigate because to show that $\varphi$ and $\psi$ are incomparable you need to show that one principle doesn't imply the other and vice versa. Something inherently more difficult. But you can still cook an infinite antichain just by considering choice from finite families. In Jech "The Axiom of Choice" the last part of Chapter 7 deals with these situation and gives a condition from which we can at least show immediately that given $p,q$ prime numbers $C_p,C_q$ are comparable if and only if $p=q$ (where $C_n$ is the axiom of choice choice from families of sets of exactly $n$ members).
There are harder things to ask, about $\sf BPI$ or about the intricate structure of the real line, or questions about regulartiy of successor cardinals, and so on and so forth. The axiom of choice can be formulated in a zillion ways, and each way can be tweaked to be weakened in a gazillion different ways. So this partial order is indeed extremely complicated (even if it's just countable).