I found the steward, but when I talked to him, it did not have the option to buy the house. Why is this? Do I have to do some sort of quest or something? I have already killed the bandit(s) but what else do I need to do? I have enough money.
Once you acquire the Dragonstone, you can purchase Breezehome for 5000 gold from Proventus Avenicci. You acquire this item during the "Bleak Falls Barrow" quest, which can be acquired from Farengar Secret-Fire once you talk to the Jarl of Whiterun.
You can also purchase this house from Brill if you have completed the "Battle for Whiterun", siding with the Stormcloaks.