Jon being King of the North: on what basis could he claim the throne? The North isn’t overthrowing monarchy/aristocracy, after all, they still all want their castles and titles, they just want to ignore Ned’s legitimate daughter in favor of his illegitimate son. What happens when it comes out he’s a Targ?

Jon is King in the North on the same grounds that Robb became King in the North – through right of acclamation as opposed to right of inheritance (since Robb’s father was not King in the North before him). It should be noted, moreover, that King in the North != Lord of Winterfell. 

As for him being illegitimate, Lady Mormont spoke for the majority: “I don’t care if he’s a bastard. Ned Stark’s blood runs through his veins.”

Him being a Targ will be a complicating factor, but when the Targaryens land on Westeros with a giant army and three dragons, having a Targ in your corner might not be a bad thing.