Jon Snow becoming the King in the North doesn't make sense in the TV series. According to the rule of succession in Winterfell, it is Sansa, a true-born Stark, not Jon, a "bastard," who is entitled to be the ruler of the north. Also, he didn't avenged the Red Wedding, like what Lord Manderly said; it was none but Sansa's (and Littlefinger's) army who struck the largest blow against the Bolton army in the Battle of the Bastards. Jon's incompentence as a leader even almost decimated the whole Stark army thanks to him, but Sansa's decision to wait out the battle before attacking the opponent ensured their win against the Boltons. Sansa is the real person who avenged the Red Wedding. Added to that, it was Lyanna Mormont, a competent woman leader, who take away Winterfell's claim from Sansa. It doesn't make sense because the show is trying to portray feminism (and it is a lot cooler to see Sansa going head to head with Cersei and Daenerys). So what gives?
In "A Storm of Swords," before dying at the Red Wedding, Robb proclaimed Jon Snow to be his successor as the King in the North, if anything happen to him. During Robb's proclamation, Maege Mormont (Lyanna's Mother) was one of the witnesses. The Mormont House, in the leadership of Lyanna, backing up Jon Snow's claim to Winterfell now makes a lot of sense.
These little but important moments in the books that were not included in the TV series made a lot of huge plot holes for the sixth season. This will not be a problem at the sixth book because George R. R. Martin, as we know, is seamless when it comes to the ongoing politics in Westeros. I think what happened really is that George only told the key moments that will happen in the sixth book without laying out to the showrunners how will it get to that point. They only know that Jon will be the King in the North at some point, but they don't know the series of events that will result to that moment.
