score:4
You're probably having trouble booking a connection through Beijing on China Eastern because they're based in Shanghai!
Instead, try Air China's online booking site: if you select origin and destination to be any cities outside China, it will offer you connections via Beijing.
If you want more creative options, see this question about finding flight connections with stopovers (TL;DR: Use ITA Matrix) and this for going about booking them (print out fare construction and go to an agent). Trying HKG X:PEK ICN
for a random date in June, I get options on Air China, China Southern, AC+Asiana, Cathay+Korean, etc. Obviously the single-airline options will have the best prices though.