ASOIAF: How did the Northerners and Riverlanders get the idea to crown Robb King in the North and Riverlands?

They were pissed off. Plain and simple. Ned had been murdered. 

They were pissed off and on top of that, they had absolutely kicked the Lannisters' ass and were ready for more.

It is undersold how dominant a position that Robb was in after the Battle of the Whispering Wood and the capture of the Kingslayer.

Tywin Lannister was literally on the run to Harrenhal, retreating so fast that his soldiers were dropping dead in the road. The Lannisters had no allies and a huge army was marching (albeit at a crawl) against them from the South. King's Landing was cut off and soon to starve. They had an idiot on the Iron Throne and Cersei as Queen Regent.

They were building a third army of untrained boys and farmers in Lannisport to replace the one that Jaime had lost.

The war to defend the Riverlands suddenly starts looking like a war of conquest, as Stannis said. The Westerlands themselves looked wide open, and indeed they would soon be invading them.

The Riverlands needed revenge. They had been badly hurt. They wanted revenge of the rape and murder of their smallfolk, the torching of their villages, the pillage of their valuables, the slaughter of their livestock and the pilfering and wanton destruction of their crops. They sure as hell didn't feel too loyal to a Throne that had condoned a war against them. 

AS for the Northmen, they were pissed off and they just got greedy. They wanted choice lands; they wanted to kick the Lannisters while they were down and dance on their faces; some of them doubtless had their eyes on the fertile and gold-filled fields of the West.

And, honestly, for all the condemnation of Robb as a politician, the Lannisters would have been utterly destroyed but for three things:

1. The unexpected betrayal of Robb by Theon and the Ironborn.
2. Stannis focusing on Renly at first rather than just taking King's Landing.
3. The conniving plan and diplomacy of Littlefinger to ally the Tyrells to their cause.
4. Edmure turning back Tywin at the fords, forcing him to go to King's Landing(and meet the Tyrells)
5. Unbelievable lucky timing in arriving to save King's Landing right as it was about to fall to Stannis.
   
   Why it's almost as if GRRM wanted Robb to fall.

Written April 21st, 2016