In published studies, mtDNA C5 (without further mutations for subclades of C5) is most common among the Paleo-Siberian Ket people near the Yenessi River, makes up about 0.1% of Koreans in published studies of Korean mtDNA, and makes up a bit larger percentage of Mongolians.
But, there are three unrelated individuals with mtDNA C5 and Korean ancestry at 23andMe (in addition to three more individuals, at least, who are related in the maternal line to one of the three).
C5c1 is found in some modern Eastern Europeans with a mutation rate based estimate of its arrival in the Neolithic era, and C5 has been found in Neolithic era mtDNA in Hungary. But, this derived clade of mtDNA C5 is not found in East Asians.
A Hypothesis Or Two
My working hypothesis is that during the relatively brief period when Korea was under Mongolian rule during the peak of the Mongol Empire in the 13th-14th century, Male Korean aristocrats were frequently required to marry Mongolian princesses by the Mongolian Empire's regime. The relationships are a plausible source for mtDNA C5 in Koreans, which would otherwise be much more rare.
The Y-DNA impact of Mongol expansion has been explored, for example, here (2003 source article here) but since these events happened in the historic era, we know that Korea was a historically fairly unique case in which introgression of Mongol elites into Korean society was female rather than male biased.
It may be possible to test this hypothesis using genological records. Korea has one of the longest time series of widespread accurate genological records found anywhere in the world (particularly with elites and the "middle class" such as it was) that goes back about about 750 years to ca. 1250 CE, just long enough to capture the arrival of Mongolian brides. It would be a painstaking, but relatively straighforward matter to identify individuals with mtDNA C5 and to trace their matriline ancestry back to this time period, although it may very well be the case that North Korean parts of these geneologies may not be available to investigators due to the near total isolation of the current North Korean state.
Another plausible source of mtDNA C5 in Koreans that could explain its rarity in published studies is that it could be more common in North Koreans than in South Koreans, since the vast majority of population genetic evidence about Korea is from North Korea. At least one of the three 23andMe clusters is specifically North Korean in origin. North Korea is, of course, geographically closer to the heartland of mtDNA C5 than South Korea, and equally important, more ecologically similar and hence more agreeable for bride exchange and migration based sources.
This could shed light on the existence of ejective glottal consonants in Korean and certain Japonic languages.