There are a number of houses you can buy in the game, but if you follow the main plot, your first opportunity will be immediately after completing the Bleak Falls Barrow quest. After speaking to the Jarl to end the quest, he'll task you with checking out a report of a dragon at a nearby tower (the next main plot quest)...and also mention that he's naming you a Thane and allowing you to buy property in the city. Before you head to the dragon (or later if you want to wait) speak to the Jarl's steward and if you have 5000 gold you can buy the house and unlock the achievement. (If you want to save your money, you can always save, buy, and reload to unlock it).