New DLCs might add on to this, but this is just my understanding of the main story so far:

(Splitting this in 3 parts cuz it's otherwise too long...)

Background:

The land of Japan was born with a great spirit stone buried deep beneath its ground. And the spirit tree grew on top of it.

Spirit stones are embodiments of prosperity, afterlife, and rebirth, taking the form of Kusanagi. But they also have the adverse effect of manifesting people's negative emotions, and so born the second spirit, Murakumo, representing the opposite as destruction.

The two are constantly clashing to maintain balance between order and chaos. 

Himiko was chosen by Kusanagi to be the queen of the new country, Yamatai.

Himiko's brother, Hiruko, wanted to harness the spirit stone's power to conquer and spread destruction. Murakumo chose him. 

Now I am not going to dig into how messy the logic can get with all the time travelling, I will assume Himiko managed to stop him by herself once which kickstarts the events afterwards, then when we came back to Antiquity, something changed, perhaps due to Kusanagi having to split half of his power to us (after all he has 2 forms, you choose one at the start, then you acquire the other half after beating Himiko) and Himiko succumbed to a corruption in this new timeline, and needed us to save her. 

Anyways, Hiruko was defeated but not entirely, and he was sealed within the spirit stone. Himiko then took up the role to be the immortal guardian of the spirit tree.

Despite being sealed, Hiruko never ceases to try to break free. And Murakumo, who represents the dark side of the spirit stone, traversed time and found our young brother Kunimatsu. Kunimatsu is based on Tokugawa Iemitsu’s real life younger brother Tokugawa Tadanaga (also known as Tokugawa Kunichiyo), who was indeed talented and favored to become shōgun by his parents, the 2nd shōgun Hidetada. But Tokugawa Ieyasu, who had officially retired but was the de-facto ruler of the shogunate, named Takechiyo to be the next (3rd) shōgun. Soon after their father's death, Iemitsu (Takechiyo) accused his brother of insanity and forced him to commit suicide. "The role of shōgun is mine... and mine alone... " is actually quite historically accurate if you choose to go with that option at the end. 

Back to Nioh 3’s story. Murakumo possessed Kunimatsu, whose pride, envy, and obsession drew the attention of Hiruko, who shared these similar emotions (pride of his own martial talents, jealousy of Himiko’s power as an Onmyo mage, and obsession to become king of the world), finally saw a chance to break free from the Great spirit stone’s seal. He took control of our young brother via Murakumo, but it was not enough. With the power of the spirit stones, he corrupted several prominent figures across different time periods, manipulating them to spread death and destruction which would enhance the power of Murakumo and his dark radiance. 

To part 2: