You need to finish the quest blood on ice. The steward will then direct you to the jarl. In turn, will ask that you'll help 5 people then you're the thane.
You have to have joined one side, and then there are different requirements. Check here for more specifics.
It didn't work
After posting this, I've managed to become thane in multiple holds without joining the war on either side, but I could not become thane of Eastmarch without taking a side, for I have already completed Blood on the Ice, and so my original advice still stands.
I've already done the neutral route: basically, you need to avoid picking a side and get through the main questline (with High Hrothgar). Eventually, you'll have to moderate a meeting between the jarls at High Hrothgar, which effectively ends the civil war questline without joining a side. After that, you can go to the jarl and he'll give you the offer to become a thane.
Well there is a second part to blood on ice where u got the wrongguy and more people die and u gotta find the right murderer, maybe that is what you need to complete to become thane.
Yeah, I just finished the Blood On Ice quest, though the steward never mentioned anything about that.
Did you choose a side in the Civil War? That's necessary for Eastmarch. In addition, you don't really talk to the steward; you talk to the jarl in order to become Thane.
Do the main quests, once you get to the mission where you ask the jarl of whiterun if you can use the reach to capture a dragon he'll tell you he can't worry about that while the war is still going on. That's when you go to the Grey beards and ask them if you can hold a council to end the war and basically make a peace treaty. Then you can go to windhelm and do as needed to become the thane and purchase property in the kingdom. You're welcome.