Unless we really stretch the idea of “starting” the Korean War, the answer to this must be Kim Il Sung, the leader of North Korea.
The Korean War started in June of 1950 when North Korea invaded South Korea. Kim Il Sung wanted to reunify the two Koreas, which had been split after World War II. He felt that his communist state was powerful enough to defeat the non-communist South Koreans. Since he ordered the invasion of the South without any provocation, we must say that Kim started the war.
Of course, the war had roots that went deeper than Kim’s decision. The base cause of the war was the enmity that arose between the communist countries and the Western democracies and their allies after WWII. The two ideologies each felt that the other was dangerous and must be destroyed. At the end of WWII, the Soviets tried to take as much of East Asia as they could. They occupied northern Korea while the US occupied southern Korea (the whole peninsula had been a Japanese colony since before the war). The US and USSR then split the peninsula into North and South Korea.
So, we can say that the antagonism between capitalism/democracy and communism started the war. However, if we are going to identify a person who started the war, it was Kim Il Sung.