This is a way to get Breezehome in Whiterun for free. This should, however, work for all homes in all cities.
You WILL still need the home cost, $5000 for Breezehome, to be able to do this. However, done correctly you will not spend it on the house.
You first need to complete the Main Quest for the Jarl of Whiterun to kill the dragon. This will net you:
As one of your rewards for this you are granted the option to speak to the Jarl’s assistant, Proventus Avenicci, to purchase a home in the city.
The home you are offered costs 5000 gold.
While you can do this at any time of the day & with any container in the keep it’s easiest done at night when Avenicci is sleeping in his quarters.
1) Wait until night when they are sleeping and then Head upstairs to the Jarl’s quarters in Dragonsreach.
2) Find Avenicci sleeping in his quarters and stand on the right side of the bed. He will be sleeping to your left w/ a wardrobe to your right.
*** 2a) Best to save at this point in case you mess up. ***
3) Speak to Avenicci and chose the option to purchase the house. Then select “I’ll Take it”.
4) Quickly push B to cancel out of the dialog with Avenicci BEFORE he gives you the house key and open up the wardrobe.
5) Store ALL of your gold in the wardrobe
6) Now let Avenicci finish his dialog to give you the house key.
7) Once you have the key, get all of your gold back out of the wardrobe.
9) Profit and enjoy your free home.
You can do this same thing for purchasing decorations for the house. You may, however, need to wait till the daytime for the decorations options to be available.
Attached is a video by VesperAzure on how to do this. There are many others if you're so inclined to search for a different one.
*No you don't need to murder the guy*