Mew cannot be caught on Pokemon Leaf Green. Players can only receive a Mew through trading with another player on either Leaf Green, Fire Red, Ruby and Sapphire.
The players wishing to trade must go to the nearest Pokemon Center. They will need to go upstairs and head to the far right to the Trade Center at the Pokemon Cable Club. Alternatively players can visit the Wireless Union Room and use the wireless adapter to connect with other devices. Once there, connect to the other player's Gameboy and begin trading.