Skyrim – How to Examine the Crime Scene During the ‘Blood on 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.

From the UESP wiki's "Blood on the Ice - Bugs" article:

Wuunferth Arrested Variation

If you speak to Wuunferth directly, instead of Jorleif, then Calixto will attempt to murder Arivanya in the Stone Quarter. If you choose to have Wuunferth arrested, then Arivanya will be found murdered. For the final showdown, a random, female NPC will be chosen as the market victim. If the game runs out of potential victims, one of two bugs will trigger:

• After speaking to Wuunferth, the journal will immediately update to stage 130. After chasing Calixto into Hjerim, he cannot be killed. He then will not leave Hjerim and becomes hostile when you enter.
  • This bug is fixed by version 1.4 of the Official Skyrim Patch.
  • (PC) To complete quest, exit Hjerim and use console commands: prid 1B11D and moveto player.
• Calixto will stay inside his House of Curiosities and will never attempt his last kill.
• If you buy Hjerim and arrest Wuunferth, when you get the objective to patrol Windhelm's streets Calixto may appear inside Hjerim, in the room behind the wardrobe, and use any equipment you may have left in it. He will be immediately aggressive and killing him advances the quest normally.

I can hit him until he falls to his knees but he gets back up and carries on.

Updating Skyrim should fix this bug. In March 2012 (PC) and April 2012 (consoles), update patch version 1.5 was released. One of its bug fixes is:

Fixed issue where Calixto would fail to die properly in Blood on the Ice.

If playing on the PC, make sure to install the Unofficial Skyrim Patch (USKP) as it fixes the bugs left unfixed in the 'Blood on the Ice' quest by the official patches. List of bug fixes by USKP related to the Blood on the Ice quest here.