1. Defeat the Elite Four and become the Pokemon Champion: This is the main requirement to access the cave.
2. Travel to the north-eastern part of the Viridian Forest: After becoming champion, you will find an entrance to Cerulean Cave within the north-eastern part of the Viridian Forest.
Important note: Cerulean Cave is a post-game area. You will only be able to enter it after you have completed the main storyline and become the Pokemon Champion.