First, your question is worded a bit poorly. The "conflict... that led to the division" of Korea was World War 2, NOT the Korean War. In August of 1945, the Soviet Union declared war on Japan and invaded Manchuria. They did it 3 months after the defeat of Germany in Europe and had amassed a decent army in the Far East. The Japanese were holding their own against the Chinese and the United States was south in Guam and Okinawa. With the nuclear attacks on Hiroshima (before the Soviet declaration) and Nagasaki (the morning of the Soviet declaration), the Japanese surrendered and the Soviets were able to drive right into Korea, stopping at the 38th parallel, a hastily decided and agreed upon line. US forces wouldn't arrive in Korea until mid-September.

With that in mind, we need to define the definition of "civil war." Part of the definition of a civil war is that it is a war occurring within a nation. In 1945, the US and the USSR administered the South and North respectively. The UN called for peninsula wide elections in 1948 to reunite the country. The Soviets, controlling a much smaller population in the North, boycotted the elections so on August 15, 1948, the Republic of Korea was founded, controlling everything from the 38th south. On September 9th, 1948, the Democratic People's Republic of Korea was founded. At that point, there were two separate, independent countries.

A couple of notes:

/u/brandonsmash mentioned Soviet military and financial assistance. The USSR, China, and North Korea were uniquely intertwined. /u/quickspore mentioned that Kim Il-sung had to travel to Moscow to get permission from Stalin, but it was a bit more complicated than that. Each player had their own motivations: Kim wanted to reunite Korea under his rule, Mao wanted to kick the United States out of Asia, and Stalin didn't care - he was focused on Europe, as was the United States. Kim's visit with Stalin "secured" Soviet air support and money, but on the condition that the Chinese commit ground forces. The Chinese had many concerns. Peng Dehuai, one of the top generals, opposed getting involved, concerned that Kim's rosy estimates of victory were wrong and that the US would use nuclear weapons against China. In the end, the Chinese intervened and Peng commanded the PVA, but the Soviet air support never flew south of Pyongyang, and the Chinese intervention was predicated on US/UN forces crossing north of the 38th parallel.

Regarding Soviet material support, they built up and outfitted the Korean People's Army but they only offered marginal air support once the Chinese army was the main force. The air support was tricky - early in the war, the MiGs were able to best F-80 Shooting Stars and F-84 Thunderjets but they were equaled by the F-86 Sabre and with superior training, the US forces were able to edge out the Soviets. The Soviet pilots flew in MiG Alley, an area along the Yalu river, north of Pyongyang. They did not want to lose a pilot over UN territory and expose their true involvement. They wore Chinese or Korean uniforms and spoke Korean over the radio. US pilots reported hearing Russian over the radios, especially swearing when a plane was shot down. The Soviets were NOT looking to expand their presence and they didn't want to give the US any excuse to expand the war, namely in Europe. It wasn't until the fall of the Soviet Union that their pilots were even recognized publicly.

To directly answer your question, was the Korean War a civil war, despite the view of an outsider that North and South Korea were separate countries, if you ask a Korean their view, most likely the answer would be yes, it was a civil war. Both North and South Korea claim the entire peninsula as their territory and they claim all people living within those borders as their citizens. It's written into both of their constitutions. During the war, Communists in the south formed irregular armies, anti-communists in the north assisted the allies, and both groups fled to their respective sides areas when the lines of battle shifted. While massive amounts of foreign troops were involved, every single Korean was involved in some way or another in the war: the entire peninsula, excluding a tiny corner in the southeast and a tiny strip in the far north, was conquered by the enemy at some point.

Check out David Halberstam's The Longest Winter for more information on the war. He has some good research from the Chinese and Soviet archives in there and a solid background on the Communist players in the war.

While it had some of the hallmarks of a civil war, there were some key factors that differentiate this from what we'd traditionally expect of a "civil war."

For one, Pyeongyang/DPRK was backed by the USSR. The Soviet involvement included military and financial assistance, and Pyeongyang sought to annex or ultimately control the RoK to the south (which was pro-west, and ultimately backed by the US).

In many senses, the aggression of the north was the first act of the Cold War and Korea was simply used as a pawn or testbed. Truman resisted his adviser's urgings to use "tactical" nuclear weapons during the conflict despite Stalin's attempts to goad the US into nuclear conflict.

If the Korean War were between two states or entities within the country it would be more akin to a civil war, but the fact that this was in many senses a political gambit on behalf of the USSR changes that dynamic entirely. Furthermore, it's not as if either side sought secession or independence; rather, Pyeongyang sought a lock on the peninsula and acted aggressively to that end.