The 14th century Persian historian Rashid al-Din is the one who describes Genghis for us. And he was described as tall with red hair and green eyes. Al-Din was writing after the Great Khan's death. But he was writing while in the court of his descendants in the Ilkhanate. And apparently both the red hair and the green eyes were family traits that a fair number of his grandchildren had.

And it isn't too implausible. Red hair and green eyes are both somewhat uncommon, but not unheard of among the Mongols and other steppe peoples. Even today modern genetic studies show that steppe peoples tend to have a mix of European and East Asian genetic markers. It shouldn't be too surprising that an area controlled by nomadic tribes had a lot of genetic wandering.

But in the end we have neither a contemporary description nor a painting. So we simply don't know precisely what Genghis looked like.

Are you sure you're not confusing him with Attila the Hun? Given his heritage Genghis Khan almost certainly had an East Asian appearance, but Attila's appearance is not quite as certain given the uncertainty surrounding the Hun/Xiongnu connection.

The incorrect blonde/blue idea could be a descendant of the Prester John myths. During one of the later crusades when the Christian European groups were head to head with the Muslims in the area they heard tales of another force sweeping in from the eastern side crushing the Muslim empire in a vice. The Christian crusaders took this as a sign of providence, and they attributed the other force incoming from the east as a lost eastern Christian kingdom ruled by Prester John.

The myth of Prester John had been building for centuries before. There are writings found here and there that talk of the disconnected, sometimes Christian, empire in the east that was some odd idealized kingdom with no crime, mythical beasts and golden cities.

When Genghis Kahn came around small reports came in that detailed a christian warlord that was beating the Muslims with maneuvers that were reminiscent of the Romans. They claimed that it must be the lost Christian nation. They thought that Prester John must be dead but the current ruler is a descendant of him, sometimes said to be his Grandson.

After this point the Mongols keep winning and eventually open up trade with the westerns, and all the while historical documents are being written that become a mix of old tale and modern truth. From this centuries later the only thing that scholars had were the writing of the previous era which were mostly the end result of a game of telephone that crossed from the steppe to Germania, with a detour in Iraq. A lot of miss truths are still floating around here and there and one describing a blonde and blue eyed king wouldn't be a surprise.

u/Quickspore quoted one of the only authentic descriptions of him from al-Din, the only other thing to add comes from the oral tales passed down on the steppe of Ghengis, and they describe his eyes as begin very much like a cats eyes, which is cool.