Instructions
Go up Nugget Bridge in Cerulean City, and then move to the left. Surf south down the river to find a cave.
Go up a winding path to the second floor and back down to the right part of the bottom floor. Surf to the southeast corner to reach the platform where Mewtwo resides.
Battle Mewtwo with a Pokemon with Water/Ice abilities. Use Articuno if you have him and try a "Blizzard" or "Ice Beam" attack to try to freeze Mewtwo. If that doesn't work, try a character that can use "Sleep" or "Paralysis" attacks against Mewtwo.
Hit Mewtwo with a variety of attacks to take away most of his hit points. You do not want to kill him; just weaken him enough to make him easy to catch.
Throw balls at Mewtwo to catch him in his weakened state. Use the Master Ball if you still have one. If not, try Great Balls.