My method....which worked for me bug free (aside from forgetting about the amulet still in my inventory):
I did the stormcloaks quest (im sure the same could be done for imperials as well as long as control is on your picked side) up before the solitude mission.
I was offered a house in the city as soon as I talked to Ulfric. I bought the house from the steward and stopped at that. (The key is just getting access to the house and from there its smooth sailing)
I went to visit my house (like normal a quest popped up to visit my new home) and blood on the ice activated automatically upon entering, this is where background knowledge of the quest helps so you know where to go and who to talk to. The quickest method and takes less then 30 mintues:
Investigate every possible spot which would include-mead bottles, chest, both wardrobes, secret room altar, pots and pans in the kitchen and the bed upstairs. Make sure to find both butcher journals 1 and 2 as well as grabbing the amulet and one of the butcher flyers.
After the above is done search out a woman named Viola in windhelm and talk to her (this is why the flyer and journals are needed). She will advise you based on your conversation to speak with the Steward....do NOT speak with the steward. Instead confront the mage accused directly and he will deny it and direct you to investigate the marketplace after dark to catch the real killer.
Wait till around 10 pm or so then head to the marketplace, you will hear a woman say help and see the killer sneaking up on her. Attack him, kill him (he may run too) and loot his body.
The only reason I did the quest was to avoid any future unencountered bugs, but from the looks of it I didn't really need to since the option to clean the house as well as getting the furniture were all available as soon as it was offered to me. One thing I made sure of was to make sure I chose clean up the mess first before buying anything then heading to my home to make sure it was clean. A bugged house will still have glass but no blood and the weird lighting conditions during the murder investigations. A clean house will obviously be fully clean no cobwebs or glass and is safe to then decorate.
I think blood on the ice is really just an addition to the house and not really a pre equisite for getting it or decorating it.