On my second playthrough I decided to complete the civil war quest first. This time I chose Imperials and Jullfric is dead. I went to the new steward and wanted to be a thane to purchase a home. Well I could purchase Hjerim right away but not improve it. I forgot about Blood on the Ice quest when purchasing it. I purchased it and it was still in disrepair.

Have no fear, head back to the steward and tell him what has transpired after investigating your home, the "search for clues" quest will remain active. The steward will direct you to a woman to talk to and show her the journal you found. The killer is later revealed as Wulferth the unliving. Head back to the steward and tell him it is the court mage Wulferth. This closes Blood on the Ice and you will now have the option to improve your home by buying the items for your home. This goes to show you can purchase the home first, then complete Blood on the Ice. Hope this helps!

That's not the killer.

The topic was more for the house. That's as far as I took it and it closed after Wulferth was revealed. Didn't go any further after that.