I'll chime in on how Japan and Korea. I've had experience with both cultures. Japan and Korea share similar pathway on how their soft power expanded particularly from economic growth and close ties to the United states.
Japan - Japan wasn't known for their products during and right after WWII. Their products would have been on the level of China today. What really got Japanese cultural popularity going was their economic growth as well as Oil prices going up during the 70s. Back then in the States big gas guzzling cars was in and little Japanese fuel efficient cars were out. That was true until the petroleum cuts and Iran's revolution and etc happening spiking gas prices up and people looking to find a way to save money on the pump. They turned to the Japanese car. This set off a wave of other Japanese products coming and an interest in Japan. With economic growth we also have culture growth as both sides were trying to understand one another . People were learning english in japan, actually this was happening long before to communicate with military personnel, and Japanese learning to get business deals going. Add to this a long line of Japanese and American military personnel intermingling and bringing back some parts of Japan back to the states you see cultural bits of Japan popping up in the states such as food(sushi, ramen, etc) and in Japan(Burgers, Pizza etc). So this was the initial start
Move to the 1980-1990s - again with the same concept of Economics, Japan was a heavy investor in the states. We could see some discomfort here from our own media(Die Hard(Nakatomi Towers), Robocop 3, etc) and events (killing of Vincent Chin because people thought he was Japanese because they thought the Japs were taking their JORBS). Essentially people who wanted to make money did what we are doing now with China, they worked with Japan a fast growing economic powerhouse. With that again came with cultural knowledge in order to work in that region.
For the millennials - Besides the cars, food, and I guess the exotic interest in their culture I'm sure most of us would agree we got into Japanese soft power from video games. Nintendo used to be a card game company and they decided to take a chance with video games. Bang Boom a few million consoles later, they had a power hold on a new form of media which was somewhat popularized by american video game companies but not the the extent that the NES did. It was cheap, yet with some amazing flagship titles that were already in the states IE Donkey Kong(which introduced mario and the princess). Through a foot hold on media this peaked interest in other forms of Japanese media in anime, and later J-pop.
Korea - Korea's story is a bit stranger, it didn't go directly to the US but through Asia, Particularly Japan. One could pinpoint the explosion starting with a K-Drama known as "Winter Solstice" in Japan. This set off a frenzy for K-Dramas and Yoon Sama was a house hold name by the early 2000s. K-Dramas were a heavy hitter and even some countries had to ban or at least set a limit to K-Dramas as their popularity was bigger then the TV shows in their own country.
K-dramas weren't the only thing growing, K-films hit the scene pretty hard with titles such as JSA, Shiri, and etc. Then with Government infusion into events such as the Pusan film festival helped establish Korea as a contender in the film market.
Korean cultural popularity I would say didn't peak until.......PSY. For real. Psy was totally unintended, he was is like the weird al of Korea. The man was known to be saying some really naughty stuff about the Korean Government if you listen to some of his older stuff. The Korean Government has been for the longest time working with their own entertainment companies to expand to other markets even giving subsidies to the music industry. The problem was K-pop in general is pretty factory made, k-pop artists one after the other in the same looks, and tone. Gets really boring after a while. Then through social media PSY made a huge hit on the scene a few months after he released gangnam style. The song was meant for Koreans as is most of the cultural foods and objects you see in the states. Watch Anthony Bourdains episode on K-Town and it gives a good explanation. Domestic Korean's didn't care if you liked our culture, it was more of this is ours we aren't catering it to you.
But like I said before economic growth did have a big factor in pushing Korean Culture to the west
Like Japan with Economic growth you see big players such as samsung, LG, etc. If you are able to have a hold on the media people use then most likely you will be able to influence the populous. Korea is the King of Cellphones atm. In both cases when either countries made a great product and economic growth happened, it sparked an international interest
TL;DR Both Japan and Korea had unique ways they got popular but starts with 1.Economic growth, 2. Exchanges in culture to work business in either countries, 3. Make a symbol that everyone seems to enjoy