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After completing the quest, talk to the Jarl in Dragonsreach. He will inform you that a home is now available for purchase, and point you towards Proventus Avenicci. You can usually find him near the throne in Dragonsreach. If he's not there, he's either in his bedroom or dining in the Great Porch. If you haven't purchased the house by the time you start the "Battle for Whiterun" quest for the Stormcloaks, you'll be required to purchase the house afterwards from Brill in Dragonsreach. Proventus will sell you the house if you can cough up the 5000 Gold payment. If you don't have the cash yet, raid a few nearby dungeons and sell the loot off to the merchants in Whiterun. Your house will be pretty bare when you first purchase it, but you can add additional furniture and decorations by purchasing them from Proventus. When you purchase additional furnishings, they will be delivered to your house automatically. You can purchase furnishings by the room. For example, furnishing your Living Room will provide 2 Weapon Racks, 1 Bookshelf, 1 Cupboard, 1 Small Table, and 2 Small Chairs. Furnishing the bedroom will provide 3 End Tables, 1 Dresser, 1 Table, 1 Chest, 2 Chairs, 1 Shield Plaque. After purchasing your home, you'll receive the key and can start using it. The house is called "Breezehome", and you can find it to the east of "Warmaiden's" just inside the western gates of Whiterun.
Talk to the Jarl. Find Proventus Avenicci in Whiterun. Purchase the house for 5000 Gold. Purchase furnishings from Proventus. Find your new house.