To get a Happiny, you have to breed a Chansey/Blissey with a Luck Incense.
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Happiny = Chansey/Blissey + Luck Incense + Ditto
Luck Incense: An item to be held by a Pokémon. It doubles a battle's prize money if the holding Pokémon joins in.
If a Pokémon that is holding this item is sent into a Trainer battle, the money earned from the battle is doubled.
If attached to a Chansey or Blissey and put in the breeding centre with a compatible father, the baby will hatch into an Happiny.
You find a Luck Incense on:
Kanto: Vermillion City
Source (1): I'm a breeder
Source (2): http://www.serebii.net/games/breed.shtml
Source (3): Bulbipedia