Korea is an interesting case, it has a large population and a reasonably easy to manage economy. However, it has little chance of becoming civilized and a huge greedy enemy to the North and a smaller but no less ferocious greedy enemy to the south.
Civilizing Korea is kind of tricky, but with luck, you can civilize before 1900s for sure.
Luck and playing your cards right will mean either success and great power status or inevitable annexation by Japan around 1900s.
Here are some tips if you want to give it a go.
Concentrate on researching industry. As soon as you get the clean coal, you should be getting a large profit. Spend all your diplomacy points on China and Japan and as soon as relations are 150+ and you have at least one tech to give to them, use it to ask for claims on Sakhalin, Amami and Okinawa(from both China and Japan). It takes 2 techs to give to each Japan and China and can be done relatively early on (1860s). As soon as you do that, claim colonies and voilà, you should be over 100 needed prestige or at least very close to it (with slave emuncipation you should be over that). This is probably the only way for Korea to get the needed 100 prestige. The only other way is trying to attack China, but it should only be done if China is at -100 prestige and you have a large enough army to quickly rush into their land and take Beijing.
If you get a chance, trade for Freedom of Trade and one of those +machine parts techs and build a bunch of cement factories everywhere. Keep doing it until you get 50 industry. That shouldn't be too hard. With a few factories, you should be making a steady profit, so spend some time calculating how much money you need to try to create the required army for 25 war score. It is not impossible!
It truly depends on luck with how quick you civilize and get the necessary warscore, since extra division events and free research point events save you a few years right away.
The fastest way to get all the war score is to wait patiently and save all the money you can (you probably need around 250 000$) and then mass produce soldiers and divisions at once. It might put you into negatives for a few years, but as long as you get the neccesary 25, you'll become civilized and see your profits triple.
As soon as you civilize, concentrate on getting industry and military research and constantly build more and more railroads and high profit factories (cement, steel, liquor). Demobilize your forces and leave just enough to be able to defend your coasts and maybe some extra for future coasts too.
Try to save up money and prepare for the war with Japan: build up your fortifications and put some military in every coastal province.
There are few things you can and probably should do now to improve your overall score:
- Buy Taiwan from China. Because it is not one of the core provinces, it'll cost you very little and you can usually just trade it for techs anyway. this will boost your population by 3+ million and if you have a full citizenship party, you can build about 5 factories there. don't forget to put some defense there too
- Build at least one high level port, so as soon as you get machine guns, you can colonize the remainder area in Java. If you're quick enough, you can get it before 1890 for sure. More colonies=more prestige. You might have to buy a few colonies from the Dutch too, so keep good relations with them.
- As soon as you've filled all your built factories and are beginning to lack manpower, you may consider attacking China. Either wait for everyone else to attack them during the Opium wars, or land your forces near Beijing, taking the capital (25% warscore) and then advance into Manchuria as soon as they move those forces to retake the capital.
- Keep in mind that Japan may strike at you at any point, so keep your mainland and Taiwan protected at all times.
- Taking Japan is usually out of the question because by the time you're ready to assault them, they usually have 100+ divisions and so unless you can establish a strong beachhead on one of the islands and hope to wait them out, you're very unlikely to succeed. China is a much easier target in my opinion.
- If you're going to fight with China or Japan, remember that if it costs you 100 prestige to declare war on them, you might want to get 100% warscore in order to at least even it out. In case of China that means taking a huge amount of land. Although in case of China, you can still get much higher overall standing despite the prestige loss if you simply build tons of factories in the conquered territory.
- If you are short on manpower, feel free to bump up your naval power. 1 Man-O-War gives just as much military score as 1 Infantry division, and with a Corvette brigade, a MoW-CC is 0.125 MIL Score, which is quite a lot. This is fairly costly and requires access to the world market for artillery, canned food and clippers, but the primary factor here is the avoidance of use of manpower.
If you're lucky enough you can become a top 3 great power before the end of the game and definitely overcome Japan.