I remember watching this for the first time in complete shock. I was really upset that Ned was gone so quick in the series. I think things went wrong for him as soon as the King demanded he become the hand. This is in part because of the Lannisters' need to be in control and Ned's inability to let things go (which isn't the worst thing). While I think the jealousy of the Lannisters was something that led to Ned's injuries (the spear through the leg), I think the ultimate demise came when he found out what John Arryn had found out about the King's bastards and Jamie's kids with his sister. He couldn't just do the hand's job and not dig into why the former hand dies; however, I think it was important for him to know why John Arryn was killed. This is a show of his chivalry (his need to understand why and how Arryn died) and how he views his friends. He keeos his friends and family safe as best he can, and I ultimately think his chivalry was his downfall.
I also think his death needed to be early because it sparks something in the Stark children. We see the shift in Sansa from this meek, docile dove into someone who is willing to do anything and everything to survive and get out of her situation with the Lannisters. Arya is all of a sudden on her own, forcing her to really grow up and think of her best interest. Robb has become the Lord of Winterfell and is in charge of the North, which is a lot of responsibility in such a short notice. Ultimately, without the head of Winterfell (Ned), the kids and Lady Stark are now on their own and without their protector. This leads to some interesting opportunities for the enemy, and also allows for the Stark children to grow in ways the audience never saw coming.