There is no fighting dojo in a cave. However the Karate King, or the master of the dojo in Saffron City, is training deep inside of Mt.Mortar.
Use the middle entrance to the cave (the one you need to Surf to), and bring Pokemon that know Waterfall, Strength, and Surf. You may want to bring some repels, as he is very far in the cave, and you will want to conserve your Pokemon's strength for the battle, as the Karate King is fairly strong. You will want to bring psychic Pokemon, as he will have a Hitmontop, Hitmonlee, and Hitmonchan, all weak against Psychic types.
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In Pokémon Gold, the fighting dojo master is originally found in Mt. Mortar. He uses a Hitmonchan and a Hitmonlee.
The Fighting Dojo Master is in Mount Mortar in the Johto Region. There are three entrances between Mahogany and Ecruteak, so take the middle entrance (and have rock climb and surf) to get to the dojo master.
He is very deep in Mt. Mortar
Beside the gym in saffron city
You can't get the badge to the fighting gym. You can however, fight the dojo master. He is in Mt. Mortar. When you beat him, he will give you a Tyrogue.