I am not an expert on the Russian Empire, but I have taken a class in graduate school on the Russian Empire, and I am very skeptical of InMooseWeTrust's answer. It may be that there weren't any formal diplomatic relations, but there are plenty of studies done on questions about nationalities within the Russian empire, and the role of Koreans in nationality policy is often discussed. The vast expanse of the Russian Empire and the continued migration of peoples, including ethnic Koreans, meant that the Tsarist state routinely interacted with ethnic Koreans, albeit on a very indirect level. To say that "nothing happened" is quite an oversimplification.

Some recommendations on where you might find an answer: The Cambridge History of Russia, Part II will give you a good introduction. I don't know if you're a college student, or if you have subscription to some database, but if you do, you should be able to access the book online.

Cambridge History of Russia, Volume II: Imperial Russia, 1689-1917. Part I, Empire. First chapter of that section by Dominic Lieven gives you a good introduction. There's also chapters from that same section that will undoubtedly help you: Chapter 2 by Theodore Weeks "Managing Empire- Tsarist Nationalities Policy" and Chapter 3 by Marc Bassin "Geographies of Imperial Identity". In this same volume, under Part III- Non-Russian Nationalities, there is Dominic Lieven's intro on nationalities policy and the Tsarist state's relationship with ethnic minorities within the Russian Empire. Admittedly, Koreans are studied less than Poles, Ukrainians, Tatars, Mongols, and Muslim minorities within the Empire.

There's also stuff on Nationalities policy in the Soviet period if you want to extend your inquiry and I know for a fact they discuss ethnic Koreans in these sources. I'll divide the articles with a forward slash. See: Terry Martin: “The Origins of Soviet Ethnic Cleansing,” J. Modern History, Dec. 1998, v. 70, no. 4, 813-61./Francine Hirsch, “Race without the Practice of Racial Politics,” Slavic Review 61, no. 1 (Spring, 2002), pp. 30-43/Terry Martin, “An Affirmative Action Empire,” in R. Suny and T. Martin, eds., A State of Nations (NY: Oxford, 2001): 67-90/Peter Gatrell, A Whole Empire Walking. Refugees in Russia during WWI (Bloomington: Indiana, 1999).

And hey, if none of that helps, there's always Wikipedia's article on ethnic Koreans in Russia to maybe get you started: Koryo-saram. https://en.wikipedia.org/wiki/Koryo-saram