This always seemed odd to me. How powerful was Soviet propaganda that it was able to convince North Koreans that they needed to fight what was essentially (and most likely literally) their family to the South? What was it that North Koreans thought they were fighting for? Was there an equivalent political conditioning by the US in South Korea?
The North didn't think of the South of their enemy, they wanted to reunify the country under their leadership. The northern leadership had their propaganda about the southern leadership and how they were American puppets, but the southern leaders had the exact same thing. Syngman Rhee wanted to attack the north to reunify the country under HIS leadership just as much as Kim Il-sung wanted to. The only difference was that the Soviets equipped the North and the Chinese pledged to support them, while the US only supplied basic, light weapons to the South out of fear that Rhee would start a war and drag the US forces into a nuclear war with the Soviets.
Also, the idea that the Korean War pitted families against each other is some poetic vision. The vast majority of Koreans (like any other nationality, really) were born, lived, and died within a very small area. There were definitely people who moved, but the majority of people who were caught up in the war were people who were fleeing one side or the other. Many young men were conscripted into whatever military was in the area so just because a person fought for the North or South doesn't mean that they had an allegiance to that side - many were just in the wrong place at the wrong time.
TLDR - both sides wanted to reunify. The war was not a war between enemy states but between ideologically different regimes sponsored by larger, ideologically different regimes in an attempt to remove the other government from power.