Well, Queen Min of the late Joseon (1890s) was pushing for modernization and ties with Russia after Japan humiliated the Qing in the Sino-Japanese War. And part of the reason the Russo-Japanese War broke out was due to the whole 'yellow peril' craze and German prodding (Wilhelm II wanted Russia to expand into Asia hoping that such expansion would alienate France and push Russia into a German alliance). In any case, the Joseon weren't staying independent at this point, Japan had been planning to take over for decades.
Well, say Alexander III dies somewhat earlier and Nicholas II and Wilhelm II get a bit cozier than OTL for various reasons. The Sino-Japanese War breaks out earlier, Queen Min is attacked but not killed (her death OTL caused quite some outrage with Japan in Korea), and King Gojong flees to the Russians as he does OTL. Out of concern of Joseon independence, Russia is granted greater influence in the Korean peninsula. With the greater economic benefits from the Far East, with both Manchuria and Korea in Russian influence, the Trans-Siberian Railroad goes a bit faster (or just have it get started earlier). Insert Boxer Rebellion equivalent due to something or another (late Qing having rebellions, easy enough to butterfly in). When the Russo-Japanese War breaks out, have Russia not be quite as incompetent (for example, no Dogger Bank incident) and the finished railroad lets them supply faster and not lose outright. Dealt a heavy enough blow to begin reforming but not lose all face, ships, etc. Also realize the utility of having Korea (a dagger pointed right at Japan's heart, as it were). With Japanese and Chinese influences curtailed, the Joseon court is now solidly Russophilic. Realizing that Japan was strong enough to challenge a European Great Power and would likely return to conquer Korea again, the Joseon, now Korean Empire, begins modernizing more energetically and embraces Russian influence and military as a means to maintain its independence, ultimately becoming a protectorate of sorts. Still wary of the British and aware of the logistical issues of absorbing over 10 million new foreign subjects on the other side of the world, Russia does not annex Korea but instead extends the Trans-Siberian railroad into Korea as far as Busan, which strengthens economic and military ties, and contributes to its modernization in an effort to create a buffer between it and Japan. Busan sees rapid development into a modern port in an effort to strengthen Russian naval power in the East as it stands closer to established Russian ports than Port Arthur (cuts down on fuel costs a bit) while giving an excellent position to threaten Japan from.
Overall, the developments cause Westminster quite some consternation as they hadn't expected Japan to win but had not expected Korea to embrace Russia so whole heartedly. Germany and Russia get along a bit better as Nicholas sees the endeavor as a success.
Or something like that. It'll be hard for Russia to take over Korea completely, due to Britain, France, and Japan desiring otherwise, but it's in their interest to prop Korea up as a bulwark against the Japanese. Plus, new markets, more resources, more prestige, etc. Late 1890s maybe?