Why can't I buy the Windhelm house (Hjerim)?

You need to start or complete the quest Blood on the Ice (In order to purchase the upgrades for the house the quest must be completed.).

Blood on the Ice is a quest initiated in Windhelm. In order to start it you must have entered and exited Windhelm at least three times.

Also the quest is automatically gained when entering the house (Hjerim) for the first time, if not previously completed.

Walkthrough

1. Approach the graveyard outside the Hall of the Dead at night; there will be several people gathered around a female corpse.
2. Speak with the guard.
3. Visit Jorleif to be granted permission to help the investigation.
4. Return to the guard.
5. Enter the Hall of the Dead and speak with Helgird.
6. Return to the body, follow the trail of blood to Hjerim.
7. Return to Jorleif for advice.
8. Pick the lock or receive a key from the Tova Shatter-Shield, Friga Shatter-Shield's mother and enter the house.
9. Investigate all of the furniture that gives you the option.The wardrobe on the north east side from entering the house will have a hidden door leading to an altar. 10.Take the Butcher's Journal #2, exit and grab the first journal from the bloody chest up against the wall, Strange Amulet and flier from the shelf (if you turn around and face the main door, the amulet is on the shelves on the right side of the door, under the fliers).
10. Visit the Candlehearth Hall or Calixto's House of Curiosities to speak with Calixto Corrium. You can show him the Strange Amulet and even sell it to him, if you so choose. He will tell you it belongs to the court mage.
11. Visit Viola Giordano. She's often found in Candlehearth Hall (note: you can't talk to her if you don't have a copy of the "Beware the Butcher" pamphlet in your inventory).

At this point, the player can complete this quest in one of two ways:

• Accuse the mage.
  Return to Jorlief to end the quest and thus accuse the court mage of the murders.

or

• Catch the killer
  1. Go and talk to Wuunferth, the court mage, rather than following the quest marker to Jorleif. Tell him you heard he was practising necromancy. He tells you he doesn't, and that the amulet and journals are not his.
  2. Go to the marketplace area in the stone quarter at night, as Wuunferth suggests, find and stop Calixto, who is about to strike again.
  3. Return to Jorleif and tell him that the killer has been caught.

When completing the quest, talking to the Steward allows the Dragonborn to buy upgrades for Hjerim. Upon becoming Thane of Eastmarch, Calder is pronounced housecarl to the Dragonborn and lodges at Hjerim.

Aditional Notes to the Quest and the House

• Purchasing the Clean-up upgrade is not necessary, but it will be unavailable after purchasing the Living Room upgrade.
• Although the Windhelm Home Decorating Guide lists a Guest Room, it cannot be purchased; it is automatically installed.
• In order to purchase Hjerim, you must have either completed the Rescue From Fort Neugrad quest if you've joined the Stormcloaks.
• If Falkreath was given to the Stormcloaks in Season Unending before you received the Rescue From Fort Neugrad quest in the Stormcloaks Civil War questline or you completed the Civil War quest line as part of the Legion,then you must complete Blood on the Ice in order to purchase the House.
• If you do not sell the amulet to Calixto and kill him as part of Blood on the Ice, it will be stuck in your inventory as a quest item. You can loot it from his body after you kill him as part of the quest.

Edit: Source is Elder Scrolls Wikia

This is a major bug, most players have encountered it, and sadly there is no known workaround that lets you buy this house or start the mission that grants you the option to buy it from he steward.

Check the Candlehearth hall, if Susanna the Wicked(the cook/waitress) is still alive, the quest hasn't activated yet. She is the one you will find dead in the graveyard.