Genghis Khan (c. 1158 – August 18, 1227), born Temüjin, was the founder and first Great Khan (Emperor) of the Mongol Empire, which became the largest contiguous empire in history after his death. After founding the Empire and being proclaimed Genghis Khan (an honorary title possibly derived from the Turkic "tengiz" — sea, meaning "the oceanic, universal ruler"), he launched the Mongol invasions that conquered most of Eurasia, reaching as far west as Poland in Europe and the Levant in the Middle East.
His descendants extended the Mongol Empire across most of Eurasia by conquering or creating vassal states in all of modern-day China, Korea, the Caucasus, Central Asia, and substantial portions of Eastern Europe and Southwest Asia.
Present-day Mongolians regard him as the founding father of Mongolia. He is also credited with bringing the Silk Road under one cohesive political environment. This brought relatively easy communication and trade between Northeast Asia, Muslim Southwest Asia, and Christian Europe, expanding the cultural horizons of all three areas.
Genghis Khan never allowed his image to be portrayed in paintings or sculptures. The only individuals to have recorded Genghis Khan's physical appearance during his lifetime were the Persian chronicler Minhaj al-Siraj Juzjani and Chinese diplomat Zhao Hong, but they don't mention red hair.
Red hair is mentioned by the Persian historian Rashid-al-Din, who wrote at the beginning of the 14th. He states that most Borjigin ancestors of Genghis Khan were "tall, long-bearded, red-haired, and bluish green-eyed," features which Genghis Khan himself had. In the Georgian Chronicles, in a passage written in the 14th century, Genghis Khan is similarly described as a large, good-looking man, with red hair. However, according to some historians, Rashid al-Din's text of red hair referred to ruddy skin complexion, and that Genghis Khan was of ruddy complexion like most of his children , except for Kublai Khan who was swarthy. (Source)
Here's a portrait of Genghis Khan's son and successor Ögedei (1186 - 1241), where he sports a reddish moustache and goatee.
However, the portrait was made after his death, during the Yuan dinasty. So, unfortunately, we don't know whether he and his father really had red hair or not, although it's interesting that such an unusual hair colour has been somewhat associated to them. It's also worth pointing out that today only a tiny minority of people belonging to the Mongolian race have facial hair.
What we know, though, is that red hair is not uncommon in Asia. For example, red hair can be found among some peoples of Afghan, Arab, Iranian, Mongolian, Turkic, Miao and Hmong descent. Ancient human remains with red and reddish-brown hair have been discovered in various parts of Asia, including the Tarim mummies of Xinjiang, China.