Following verbs of perception (hear, see, etc.) you may use either the infinitive form or the gerund form. The grammar and meaning are different for each.
In your example the infinitive is teach or to teach. So the correct grammar is:
I would like to hear our king teach me.
Alternately, if you want to express an ongoing action, use the gerund:
I would like to hear our king teaching me.
Now ... while this is correct grammar this is not natural English. For starters, hear is passive while listen is active. If the king were teaching you, you would probably pay attention, so the more appropriate verb would be to listen.
But again, it's odd to say you would only listen to someone teach you something. Usually we would get (or be given) a lesson, or attend a lesson, or at the very least (if we were not part of the class) observe a lesson. We would get the full experience, not just the auditory part. So if you are the one being taught by the king, you would use a more comprehensive verb to describe the experience, and not just "hear":
I would like to have our king teach me.
Since this is an odd example, here are some others:
I would like to listen to him play Beethoven.
I would like to listen to him playing Beethoven.
I would like to watch her cook us dinner.
I would like to watch her cooking us dinner.
I love to hear the birds sing
I love to hear the birds singing
More on verbs of perception