Mewtwo can only be found within the depths of the Cerulean Cave. Mewtwo is incredibly difficult to defeat if you are not well prepared. First, make sure you have caught at least 40 Pokemon prior to entering the cave. You will need multiple strong, high level Pokemon to defeat the various Pokemon inside the cave. Also, be sure to stock up on plenty of Hyper and Max Potions.
To reach the Cerulean Cave, first visit the Pokemon Center on Route 24 right above Fuchsia City. Speak to the woman sitting at the table to your right as soon as you enter and she will ask if you want to view a slide show about caves. Answer “yes” and she will show you several slides. After the slide show, she will give you a Town Map depicting the location of the Cerulean Cave. Take this map and show it to the police officer to the left of the center. After showing him your map, the officer will let you pass. Head up to the Route 25, which is the gate above Route 24.
Surf across the water and you will reach a small island with another officer sitting on the left. He will not let you pass until you have earned eight gym badges. After obtaining the 8th gym badge, head back to Route 25. The Officer will move and you can now continue on to the Cerulean Cave.
Before entering the Cerulean cave, prepare your team. Make sure you have a Pokemon that knows Surf and a Pokemon that can learn Flash (such as a Magnezone). When you are ready, enter the Cerulean Cave and use Flash. Begin running forward along the path while avoiding the wild Pokemon that appear periodically. Continue until you reach a pond; use Surf to traverse the pond and land along the shore at the other side.
Continue along this path until you come to a fork. Take the left path and walk further until you come across another fork with two paths. Take the right path and continue until you reach another pond. Surf the pond and land on the shore at the other side. As soon as you make land, look down and you will see Mewtwo in a pit beneath your feet. If you wish to catch Mewtwo, make sure to lower its HP without defeating it. When its health is low enough, throw an Ultra or better ball to catch it.
If you defeat Mewtwo, it will be gone forever. Mewtwo will not reappear within the Cerulean Cave nor can you find it elsewhere.