More specifically from Busan (but other sites also possible).
My plan is to do Japan - South Korea - Hong Kong by sea.
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There are a number of ferries between South Korea and China, via Tianjin or Qingdao. Both South Korea and China have a well-connected railway network, so it shouldn't be a problem to travel onward by train from the arrival point in China to Hong Kong.
For what it's worth, the same page also lists ferries between South Korea and Japan.
South Korea to Hong Kong is quite far, so I would be surprised if there was any direct ferry. One might hope to travel via Taiwan, but unfortunately, there is no longer any ferry connecting Japan and Taiwan. However, see hippietrails comment below and this answer for more information.
Another question on this site actually indirectly answers this. Although there are no scheduled ferries between Hong Kong and Korea, you can take a 'freighter cruise' between the them:
Considering that you are traveling on a freighter, it may not fit your travel plans however it does seem possible.
Just to answer a specific point from your question, there are currently no ferries from Busan to China. The only ferries currently from Busan are to various Japanese ports.
Building up on one of Andrew's answers (+1), there are many freighters on this route and some of them do take passengers. There are a number of agents selling these trips and while they often focus on long transcontinental trips, it's sometimes also possible to book “partial journeys” between two ports on a longer freighter line.
As an example, this German agency lets you pick your starting point and destination from the list (“Teilstrecken möglich” means something like “partial route possible”). It's not especially cheap however, a trip from Busan to Hong Kong would cost about EUR 2300.