The Annals of Joseon Dynasty (조선왕조실록) has one prominent bisexual individual- the Crown Princess Sunbin Bong-ssi (1414-1436).

She was the wife of the Crown Prince Hyang in Joseon. Crown Prince Hyang is the son of King Sejong, the guy who created Hangeul (Korean alphabets).

When the homosexual encounter was discovered, it was a huge scandal. An investigation that followed revealed that Bong-ssi discovered that her maid So-ssang was in a homosexual relationship with Dan-ji, another maid, and used her position of power to coerce So-ssang into having sex with her.

The records say that the maid often complained to her peers that 'the crown princess loves me like no other, and it frightens me so'.

She was kicked out, and died on the same year.

Given other records on her person as being highly unstable (there's a whole list of deviant behaviours written long before the actual homosexual scandal erupted), it is speculated that she committed suicide.

The Emperor who called himself Queen

I'm not sure if homosexual is exactly the right phrase, but that is a more modern term. Nevertheless this is one of my favorite Roman emperors, not because of greatness or anything, but just straight up weirdness.

Emperor Elagabalus, Roman emperor from 218 - 222, would probably fall under your question. In Ancient Rome homosexuality itself would not have particularly been a problem, though men were expected to marry and produce heirs, rather the question would have been who was the dominant partner.

For instance, according to Suetonius, when Julius Caesar was a young man he was sent to the court of Nicomedes as an ambassador, to raise a fleet with the king's help. But he stayed so long with the king that later detractors would insult him by calling him the Queen of Bithynia.

Apparently the more submissive role was the insult, not so much the sexual act itself.

So now we fast forward back to Elagabalus. He was raised in Syria and so tended to follow more eastern customs. He was said to use make up, for instance and dress in a more eastern style. The elite of Rome didn't care for his style from the start, he would be considered flamboyant, to say the least.

As to the question of sexuality though, again I'm not sure that modern term exactly fits. He was married to several women, though this may have been to make himself feel more like a woman, and to learn how to treat his male lovers. He even forced, himself on a vestal virgin. This is incredibly taboo as you might imagine.
The thing is, he didn't stop there. According to Herodian he actually married a god.

"Not content with making a mockery of human marriage, he even sought a wife for the god whose priest he was. He brought into his own bedroom the statue of Pallas which the Romans worship hidden and unseen. Even though this statue had not been moved from the time when it was first brought from Troy, except when the temple of Vesta was destroyed by fire, Heliogabalus moved it now and brought it into the palace to be married to his god."

But your question was about homosexuality, so, yeah, he also took men as lovers. Cassius Dio writes that he actually prostituted himself out to men.

"He used his body both for doing and allowing many strange things, which no one could endure to tell or hear of; but his most conspicuous acts, which it would be impossible to conceal, were the following. He would go to the taverns by night, wearing a wig, and there ply the trade of a female huckster. He frequented the notorious brothels, drove out the prostitutes, and played the prostitute himself. Finally, he set aside a room in the palace and there committed his indecencies, always standing nude at the door of the room, as the harlots do, and shaking the curtain which hung from gold rings, while in a soft and melting voice he solicited the passers-by. There were, of course, men who had been specially instructed to play their part."

He often boasted about how many lovers he had, but he did have a favorite it seems. His favorite charioteer, Hierocles. He went so far as to call himself Hierocles' Queen, and may have, again according Cassius Dio, actually married the young man.

He is even said to have given special honors to a man rumored to have particularly large, uhm, parts. Sending an entourage out to fetch him the man greeted the emperor with something like "Hail my Lord Emperor." To which the emperor replied, "Do not call me Lord, for I am a Lady. Then took the man to his bath.

So, yeah, you gotta love this guy. He became emperor and just ran out of fucks to give. It didn't last too long though. This sort of lifestyle did not sit well with the Praetorian Guard, and those were the people who kept the Emperor, well, alive. Eventually it got to be too much and the guard demanded, (yeah, the Guard could actually demand the emperor do things because, the kept him alive, and stuff), that he come to their camp with his cousin, a boy he had named as successor but then changed his mind when he saw how popular he was becoming.

They cheered the boy and ignored the Emperor. He called for all such insurrectionists to be arrested on the spot. The Guard said, yeah, no. And promptly beheaded him.

Thus died Elagabalus, Emperor of Rome and Hierocles' Queen.

EDIT some spelling and formatting but mostly I thought up a cool title