Mt. Mortar after winning at Blackthorn Gym. You will need a Pokemon that knows waterfall and an open space in your party because he will give you a Tyrogue after you beat him in a Pokemon battle.
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The Karate King can be found in Mt.Mortar in Pokemon Gold. You will need the HM moves Surf and Waterfall to reach the part of the cave where the Karate King is training.
you have to get to the fighting dojo in saffaron on kanto
In Pokemon Gold, Silver, Crystal, Heartgold, and Soulsilver the Karate King is in Mt. Mortar, East of Ekruteak City. He will have a Hitmonlee and Hitmonchan, and give a Tyrogue upon defeat if you accept it and have a space empty.
The Karate King cannot be returned to his Gym in Pokémon Gold since it has been disbanded story-wise. You can nevertheless find the Karate King in Mt. Motor to battle and receive a Pokémon from. Make sure you have an empty space in your party or you won't receive the Pokémon.
The last badge is off the coast of cinnabar it is the red dot in the left corner of the map