The concept of a Penguin King is pretty interesting and is executed very well and very greatly.
He is shown to care about the Snow Kingdom and the Penguins, such as the beginning scene of him and the Penguins.
He has a bunch of hilarious moments such as:
The opening scene where he and the Penguins pathetically try to defeat Bowser and his army by throwing Snowballs at him, Kamek teleporting, hitting one of the Koopa Soldiers, and getting captured by Bowser.
The Penguin King telling the Lumalee that the prisoners are already depressed enough.
Him telling Lumalee to shut up.
He gets rescued by Tanooki Mario and it is implied that after Bowser got defeated, he went back to his kingdom and he and the penguins rebuilt their kingdom.
As a supporting character, he makes a lot of impact on the film.
Khary Payton does a great job voicing him thanks to him using his normal-sounding voice he uses in most of his media such as Teen Titans Go! (coincidentally Scott Menville who voices as Robin in Teen Titans Go! starred as the voice of Koopa General).
His design is very cool and nice.
He has a bunch of hilarious/memorable lines due to the writing such as:
"That is but a taste of our fury. Do you yield?"
"Pay him no heed. He's cute, but he is (cuckoo noise)."
"Please! We are depressed enough!"
"Oh, you've got to be kidding me!"
"SHUT UP, SHUT UP, SHUT UP!"
Due to the film's pacing being way too fast, he and his Penguins are never shown going back to the Snow Kingdom.
Him and the Penguins getting burned in Lava is a pretty disturbing concept for Mario standards.
He doesn't really do anything when he and the Peguins are in their cages until Tanooki Mario rescues him.
Plot hole: For some reason, the Penguin King and the Peguins don't the Super Star to become invincible in the beginning when Bowser, Kamek, and the Koopa General are taking over the Snow Kingdom to stop Bowser and his troops, which is odd. Though this also can be excused by the fact that they didn't know it's use.
While he does get kidnapped, it was huge missed opportunity to have him and all of the other prisoners and maybe the Lumalee escape from their cages with help from the other prisoners, even if he ended up getting caught and being put back in his cage, and there could have been more scenes of him and the Penguins, and the fact that Luigi got massive character development, and he doesn't really do anything when he is in his cage waiting for Tanooki Mario. This would've given the Penguin King more scenes and character development. He doesn't even get that much screen time when he is in the cage for a while. According to concept art, the Penguin King was supposed to do a breakout in his cage.