This pairing is usually referred to as "Kimchiburger", taken from the fact that America enjoys his burgers, while Korea enjoys kimchi.
Though they haven't interacted much in the webcomics, the pairing comes from their history. America had been with Korea through hard times, for example, the Korean War (also known as the Forgotten War). The only time they had shown to actually be interacting in the webcomics was during a meeting, where Korea disagreed with Japan while agreeing with America, though America and Japan had already agreed.
America and Korea were also both in the "Do your best! Japanese snacks!" strip, but they did not interact. And Korea mentioned America in the "Breasts are Forever" strip, when Japan turns down Korea's request to touch his chest and Korea declairs that "If you don't let me touch them, I'll make a movie... and then show it to America later..." It isn't clear why he'd want to watch it with America, but one can assume it is because of their historically close friendship.