Instructions
Capture a female Marill and a male Marill at Mount Mortar. If you walk in the long grass, you have a one percent chance of finding a Marill. Listen to the radio station called "Mary and Oak" on your PokeGear. If Marill is the Swarm Pokemon of the day, 40 percent of the wild Pokemon at Mount Mortar will be Marill.
Go to Cerulean Cave, Kanto and head to the northern edge on Floor 1. Pick up the Item ball, which contains the Sea Incense. Your character will place the Sea Incense in the Items pocket of your Bag. From the Items pocket, select and give the Sea Incense to the female Marill.
Visit the Pokemon Daycare on Route 34. Give the female and male Marill to the woman behind the counter. Leave the building and bike through Route 34 and Goldenrod City. Wait for the Daycare Man, who is standing outside of the Pokemon Daycare, to give you a call. Speak to him and he will give you an egg.
Take the female and male Marill out of the Pokemon Daycare, unless you want more than one Azurill. Put a Pokemon with the "Flame Body" or "Magma Armour" ability in your party. If you have a Pokemon with either ability, the Azurill egg will hatch in about 1,280 steps. Otherwise, the Azurill egg will hatch in about 2,560 steps.
Cycle through Goldenrod City and Route 34. Since cycling is faster than walking, you will take more steps in less time by cycling. Therefore, you will hatch the Azurill egg faster by riding your bike. After about 1,280 or 2,560 steps, the Pokemon egg will hatch into Azurill.