I'd say his main problem is wanting to be recognised for something he's not, rather than what he is. He wants respect. This obsession comes from his looks. He was born a freak and was teased for it. So living amongst the upper class folk, where looks aren't important and money is the main reason why people look at him, that's Penguin's haven.
I wouldn't class him as insane, but he has definitely violent tendencies. That obviously generated from his childhood. All that bitterness and anger at being bullied for so long manifested into violence.
His criminal behaviour is at best extreme, but can't be classed with insanity. It comes from his longing to be somebody else. Someone who commands power and respect from being in high class. He's more of a pitiful character.
He's genuinely not a evil person, and he's not out to hurt anyone.
His rush with wanting to be top dog in Gotham obviously comes from his desire and obsession to be respected. Penguin wants to be liked for what he has.
Now we can tell, he strongly wants to be liked by people. He's still that kid with no friends. Wanting to be accepted by everyone, but never had anything going for him. So he steals and cheats his way into high society, just so he will be liked by people. Receiving the attention, the compliments, the atmosphere that he never had when he was young. He did it it all, just so he could 'fit in'.
Being Gotham's leading mob boss is him being too high and cocky with his life in upper class. Not content with being the most popular person in a special club, he wants it by being a criminal too. So Penguin is high on his own power. His desire to be rich and powerful influences his continuing path of being a criminal. All because of him wanting to be liked and not bullied.