As others have pointed out, he told Benjen..First Ranger. Did Benjen care that Ned killed the boy? No not at all, because that's what's expected.
And Jaime looked like a right bastard when they found him, we know he isn't because we have all the facts. If I was in Ned's situation, having seen what Tywin did to King's Landing, having seen what was done to Elia, Rhaenys and Aegon, I'd be just as distrustful and pissed off as Ned was. It looked like Jaime was in on the betrayal, and at that they don't get the ****ing benefit of the doubt.
Ned's not a perfect man, he was flawed like every other character in the series, but every choice he made in his life was based on his own moral code of ethics. There's a reason why the North Remembers.