Yes it means someone is dishonest, corrupt or immoral. Instead of saying “working his sleaze” a more common use of the word is saying someone or something is “sleazy”. “I don’t like this club, it’s pretty sleazy.” “She’s really sleazy, I would stay away from her if I were you.”

In simple terms it means he is dishonestly furthering his own agenda through corrupt behavior.

Sleaze is corrupt or immoral behaviour, so it means the character is acting in a corrupt or immoral way.