There seems little doubt that Kim Il-sung as head of the DPRK actually ordered the invasion of the south on June 25, 1950, and we should attempt to understand the factors that led him and his colleagues to make that decision.
The USA’s government is reasonably open, it may not always seem transparent, but in time most archival information is released to the public and is available to scholars. That was not the case with either the DPRK not the PRC at that time, not the USSR of the time period, But since the collapse of the USSR, much more archival information has become available from the extant Soviet Archives in contemporary Russia. It is possible to trace the conversations that discussed the DPRK’s belief that they could invade the spout, and most importantly why Kim Il-sung believed he would win and so re-unite all of Korea.
By early 1950 the north had a reasonably well equipped army, mostly supplied by the USSR, probably from the large pool of Soviet equipment left over after the Soviet’s successful (and very short) war against Japan. Moreover in late 1949, tens of thousands of well-trained and experienced Koreans, who had been fighting in the Chinese civil war between the GMD and the CPC, were able to return to Korea. Naturally, and irrespective of their original home area in Korea, they chose to return to the DPRK. Their leadership were likely to be gung-ho about the chances of victory over the south.
We should balance the DPRK’s attitude with that of Syngman Rhee’s administration and his generals. Both Rhee and his generals spoke openly about re-taking the north by force. Most of the south’s army leadership had been trained in the Japanese Army and were likely quite confident in their ability to win a war with the north. And, many of the brief military conflicts (incursions into the north) that had been occurring along the border are believed to have been initiated by the south In that kind of atmosphere both sides likely believed that one day, the other side, would make a genuine invasion attempt.
But, perhaps an overlooked, but very likely important factor, is the fact that between the end of 1948 and June 1950 South Korea itself had experienced a violent and bloody guerilla revolt/war by native south Korean leftists on Cheju Island and in Yosu-Sunch’on in South Cholla. The revolt was unsuccessful, but we can consider that the DPRK government could have easily believed that in the ROK, there were large numbers of leftist thinking Koreans that would rise up in revolt once and invasion commenced.
So irrespective of whether or not either the USSR or the PRC actually approved the invasion, (although it seems certain that neither Stalin or the CPC would have said No!), Kim Il-sung seems to have believed he could win and that he should make the first move, rather than letting the south make the first move.
And, Kim was likely feeling he’d made the right decision as the north’ blitzkrieg attack overwhelmed the ROK army, and in three days took Seoul, and by early August had the ROK military confined to a small pocket of land around Busan.
Any elation was however short-lived. America’s MacArthur and his quite brilliant decision to make an amphibious assault on Inchon (near Seoul) would have ended the north’s hopes of an early and total victory. By the end of September the combined western and ROK armies were once again in control of Seoul, and, because of (I think) MacArthur’s quite understandable decision to continue heading north saw his forces coming close to the border with China, which in turn was likely to have been causing some late nights in Beijing, where the leadership would have been pondering American intentions.