Since you haven't taken sides in the civil war, these do not apply yet.

It is possible that after the Battle of Windhelm, Brunwulf Free-Winter may not appear in the Palace of the Kings, preventing you from being named Thane of Eastmarch and preventing the purchase of Hjerim.The option to buy the house may exist but you are unable to actually purchase it even after completing the prerequisites as the dialogue still says it is unable to be purchased due to unpleasantness.Note also that there are a couple known bugs when trying to purchase Hjerim:

Thus, you could get the tiny quests out of the way before beginning the civil war. I believe you can complete five quests helping NPCs in Eastmarch before even asking about becoming thane, since I've done something similar in other holds.

To become thane, you must join the civil war and complete one of the quests (Liberate Falkreath for Ulfric or conquer Windhelm and depose him.) You must then complete the five quests to help the people of Eastmarch, after which you can buy the house and become thane. In your case, you may have completed Blood on the Ice (but check the spoilers for that quest to be sure.) But this quest is not necessary either for purchasing the house or for becoming Thane of Eastmarch. There's an answer to this question: Why can't I buy the Windhelm house (Hjerim)?.