How can I start the Blood on the Ice quest?

There are a few possibilities here, according to the uesp entry on the quest

The trigger may take some time as you'll have to wait for the woman to be murdered. Personally I came across the scene in the day time while passing through the area, I did not wait in the graveyard (that may cause the murder to not trigger).

You can also try to trigger the quest by purchasing the house from Jorleif during the stormcloak rebellion quest line.

However, it is known (see the glitches section in the entry) that the quest is bugged and may not start at all, not even via the console. In this case then you are out of luck.

This quest also didn't trigger for me.

To force it, you can pickpocket a Hjerim Key from someone in town (I don't remember who has it, but there are several) and enter the house. (Tova has one.)

However, the quest still bugged for me later -- so you may still need to use the console at some point.

Starting "Blood on the Ice"

Make sure you meet the requirements needed to start the Blood on the Ice 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.

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.

Official and Unofficial Patch Fixes

The Unofficial Skyrim Patch (USKP) (original / Legendary Edition - PC) / (Special Edition - PC) / (PS4) / (Xbox One) , along with the official patches should now fix most of the bugs associated with the Blood on the Ice quest.

Official patch changes:

• Killing Viola before or after Blood on the Ice no longer blocks progression.
• Fixed issue where Calixto would fail to die properly in Blood on the Ice.
• Fixed numerous issues with Blood on the Ice not triggering properly
• In Blood on the Ice, Calixto can now be killed if player owns a house in Windhelm

Unofficial Skyrim Patch changes:

• Blood on the Ice will now allow you to complete the quest after the Imperials have taken over Windhelm. The guard acting as a CSI will switch properly to an Imperial as needed.
• In "Blood on the Ice" (MS11), the player's chest in Hjerim would not enable unless Wuunferth was arrested for the crime.
• Viola and Arivanya are both required NPCs for "Blood on the Ice" (MS11) but were not properly added to the pre-quest that holds their essential aliases. If they die, MS11 will not start. If they are dead and MS11 has not yet started, they will be retroactively resurrected.
• In MS11, the objective "Get Assistance from Jorleif" is never cleared during the quest.
• MS11 was never counted toward the "Sideways" Steam achievement.
• 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.
• If the player chooses not to sell the amulet to Claxito during MS11, they will have it identified when Wuunferth realizes what it is.
• If MS11 has already past the stage where it could be identified and the player is still carrying around the Strange Amulet, it will be swapped out for the Necromancer's Amulet they should have received.
• MS11 would fail to start if the dead body in the Hall of the Dead in Windhelm had been purged by the corpse cleanup code. A new replacement body is generated to correct this as well as making the body's alias optional so the quest will start.
• MS11SecretDoorActivatorScript has been fixed to keep it from trying to run Blood on the Ice content when that quest is not running.
• The Strange Amulet (D2328) will no longer remain as an undroppable quest item when the Blood on the Ice quest is completed.

^{USKP full version history}
Bug notes from the UESP wiki, Blood on the Ice - Bugs article:

• Even before the quest is initiated, speaking to Calixto outside his store will cause him to close it permanently. Do not attempt to pick the lock, as he will become hostile. Instead, if you want information about the Amulet, find him at Candlehearth Hall between 5pm and 7pm.
• When you are told to question the witnesses, stay away from Hjerim. Approaching the front door will trigger the 'Gain access to Hjerim' update and clear the graveyard. This will leave you with no witnesses or map markers for the remainder of the quest.
• If the quest is started and the player waits too long before continuing it, the blood splatters leading to Hjerim can disappear.
• If you take the first journal to Viola, she will direct you to talk to Jorleif whether or not you have found the amulet. The quest will proceed as though you had found the amulet when talking to Jorleif and Wuunferth, even though you do not.
• Sometimes, if you cause Wuunferth to be arrested, when you return to Windhelm, Arivanya will have been killed, but no guard will be standing by the body to re-activate the quest, rendering it unfinishable. In this case the guard is standing in town, but in the outside world. You must exit Winterhold, then use the console command tlc to fly into the town without entering the gate. The guard will be standing relatively at the same place.