I know we have the consistency of $\mathsf{ZF} + \mathsf{AC} + \mathsf{GCH}$, $\mathsf{ZF} + \neg \mathsf{AC}$, and $\mathsf{ZF} + \mathsf{AC} + \neg \mathsf{GCH}$.
What about $\mathsf{ZF} + \neg \mathsf{AC} + \mathsf{CH}$ and $\mathsf{ZF} + \neg\mathsf{AC} + \neg\mathsf{CH}$?
There's really nothing much to it.
We can easily use symmetric forcing the axiom of choice to fail only above rank $\omega+\omega$. Then if $\sf CH$ was true in the ground model, it will be true while $\sf AC$ is false; and if $\sf CH$ was false in the ground model, it will be false in the symmetric extension.
Do note, however, that $\sf CH$ has several formulations which end up non-equivalent when the axiom of choice fails. See How to formulate continuum hypothesis without the axiom of choice? for more details. And while we're handing out links, The Continuum Hypothesis & The Axiom of Choice seems relevant as well.