I'll start with playerwise, and move on to champwise afterwards, since the two are pretty intertwined.
To me, a jungler's duties are as follows.
A) Securing global objectives (dragon early on, baron later).
B) Preventing the enemy's jungler from getting anything done.
C) Other crap (ganking, farming, etc.)
The perfect jungler can predict their opponent's movements and always be there to countergank. They can find and abuse every opportunity to get vision in the enemy's jungle and steal camps away. The other jungler should feel powerless, because everything they attempt has been predicted and countered by their opponent. The perfect jungler can win every lane without having to show themselves because they can nullify any chance of the opponent ganking while keeping open the possibility of ganking themselves.
To me, KaKAO is the best jungler in the world. Nobody can match the early pressure he brings to the table. The series between the KTA and NWS exemplifies what makes him so great, in my opinion - playing both sides of the Elise vs. Nocturne matchup in two games, he first invades Watch as Elise and solo kills him, then in the next game baits Watch into invading him and solo kills him again.
Knowing my values as a jungler - I think that pre-nerf Evelynn was the closest thing to a perfect jungler that I've ever seen. The early pressure she could bring without even showing on the map, the vision control she could secure, and her ability to invade the enemy's jungle and kill nearly any champion 1v1 early on really exemplify what I feel a jungler should do. Her and Elise were my favorite champions by far before their nerfs, and even after in Evelynn's case.