Starting "Blood on the Ice"
(Note: If you've already finished "Blood on the Ice" skip this section and scroll further down below.)
You need to finish the Blood on the Ice quest first before the steward will give you the conversation option to buy the Windhelm house (Hjerim). Make sure you meet the requirements needed to start that quest:
"This quest is triggered by a counter that starts at the beginning of
the game which will increase each time you visit the city of Windhelm
or the immediate surrounding area. When it reaches four, the Blood on
the Ice quest will be available, although two more conditions must be
met:
- The player has to enter Windhelm, either through the main gate, the docks gate, or by fast-traveling, between 7pm and 7am. The graveyard
scene should then be triggered and the quest ready to go.
- Tova Shatter-Shield and Arivanya need to be alive (see bugs section).
The easiest way to trigger the quest is to walk back and forth between
Windhelm Stables and Brandy-Mug Farm two-three times, then
enter Windhelm between 7pm and 7am. If the graveyard scene is not
triggered by this, there's little hope you can trigger the quest at
all.
NOTE: You can always check whether or not you have met the
requirements for the quest to start, as the lock to Hjerim will change
from a “key required”-lock to a normal lock when the quest becomes
available.
If Tova and Arivanya are alive, the quest has not yet initiated
because the counter is less than four, and you obtain a key to Hjerim,
then you can trigger this quest by entering Hjerim with that key and
picking up one of the Beware the Butcher! pamphlets. However, the
quest will not start in the same way as outlined in the walkthrough
above; you will have to figure out what to do on your own without help
from Jorleif, however all of the same options will be open to you
except that none of the witnesses will be present in the graveyard;
you can sell Calixto the Strange Amulet, you can have Wuunferth
arrested or talk to him to catch Calixto, and you will have to talk to
Viola in order to continue on with the quest after picking up a
pamphlet."
Source: UESP wiki, "Blood on the Ice - How to Start the Quest"
Possible conflicts with the Dark Brotherhood quest, "Mourning Never Comes"
Although most of the NPCs needed for the quest to initiate are marked as essential, another quest, namely the Dark Brotherhood quest Mourning Never Comes, will encourage you to kill Nilsine Shatter-Shield. Even though she is not related to this quest, her mother, Tova Shatter-Shield, will commit suicide if Nilsine is killed, and will therefore render this quest impossible to initiate.
Source: UESP wiki's "Blood on the Ice - Bugs" article
A possible fix, from the UESP wiki's "Blood on the Ice - Notes" article:
If you've killed Nilsine in order to get a bonus reward in the quest Mourning Never Comes, Tova will be dead inside her house, and you will need to pick the lock of her house in order to get the key to Hjerim. This will only happen if you met the requirements for Blood on the Ice and then completed Mourning Never Comes.
Also, if you are playing on the PC, install the Unofficial Skyrim Patch to fix a lot of bugs related to Blood on the Ice quest - one of which is related to the key to Hjerim:
If Tova Shatter-Shield is dead (actually just disabled, but that's another story) because you completed "Mourning Never Comes" (DB03) before MS11, a substitute key for Hjerim will be made available by the USKP.
Already finished "Blood on the Ice"
If you've already finished Blood on the Ice and the steward is still not giving the conversation options for inquiring about or buying the house, then you might have encountered a bug. If you're playing on the PC, the console command setstage 000A7B33 10
should allow the steward to give the conversation options necessary for you to buy the house. (As suggested by the UESP wiki's Hjerim - Notes article.) This worked for one of my characters which has encountered that bug.
If you're playing on the PC, make sure you also have the Unofficial Skyrim Patch installed as it fixes a lot of bugs related to the Blood on the Ice quest.