There's lots of characters there...but I will do my best to be succinct. I will choose some but not everyone. Sorry.
Minister of the Right (Grandfather to the Emperor's firstborn son, father of Lady of Kokiden Pavillion.) Doesn't have good relationship with Minister of the Left. At one point he offered his fourth and favorite daughter to Genji.
Lady of Paulownia Court's Parents: The father was a grand councilor who has passed away, and the mother comes from a good lineage. Both of them loved Lady of Paulownia Court greatly despite their lack of strong backing. The father also dreamed of sending the girl to court no matter what, the desire was fulfilled. For a while the mother raised Genji but then she sends him back and soon passes away.
Emperor (Genji's Father): He is deeply in love with Lady of Paulownia Court and unfortunately his love couldn't help her continue to live. He desired for Genji to become a crown prince but was unable to do so and has no choice but to let Lady of Kokiden Pavilion's son become the crown prince. Instead he designated Genji as a peasant. He loves Genji greatly and does everything he can for him. Later on he finds a woman that resembles Lady of Paulownia Court, named Fujitsubo and marries her. In later chapters, Fujitsubo bears him a son who is actually his grandchild by Genji.
Lady of Paulownia Court (Genji's Mother): She comes from a family without a strong backing and has captivated the Emperor's heart. She was bullied by other Emperor's wives and after giving birth to Genji, she passes away.
Lady of Kokiden Pavilion: Lady of Kokiden Pavillion is the main of Emperor's wives and she is also the most jealous. She is afraid that her son will lose his status due to Genji, and she is also a chief bully of Lady of Paulownia Court. She also has two daughters and she also looked after Genji a little before Fujitsubo came into the picture. The emperor couldn't ignore her. Her son is the crown prince.
Prince Hyobu (Fujitsubo's elder brother): He is the father to Murasaki and elder brother to Fujitsubo. He is married to a woman who caused Murasaki's mother to go into decline and at one point wanted to raise Murasaki, although Genji snatched her away. He and Fujitsubo are the children of the former empress. He is described as elegant and if he were a woman, then Genji would pursue him.
Fujitsubo: She is a Fourth Princess, daughter of the emperor before Genji's father. She greatly resembles Genji's mother, thus the emperor decides to marry her. She is also an orphan. Genji constantly sees her and has fallen in love with her. She is called "Lady of the Radiant Sun." She becomes pregnant with Genji's child without the emperor finding out. She gives birth to a prince and is made empress. Later on after the Emperor dies she becomes a nun.
Minister of the Left (Father to Genji's wife) and his wife: He bestowed the official cap on Genji and has only one daughter. The mother was a princess of the blood, and was emperor's sister. Doesn't have good relationship with Minister of the Right and has multiple children, one of them a son who is guards lieutenant. In chapter 10, he resigns due to Minister of the Right's influence. In chapter 19, he passes away.
Genji: He is the protagonist of the book who seems to succeed in almost everything he does that relates to government and so forth. He has many adventures and is beloved and favorite of his father the emperor. He is described as handsome and intelligent and not long for the world. In later chapters he has three children; two sons and a daughter. Perhaps more than 70 percent of the book is of his adventures. He is also known as "The Shining One." By Chapter 42 he has passed away at an unknown age, (possibly in mid 50s?)
The Suzaku Emperor: His mother is the infamous Kokiden and he is the crown prince, to rule after Genji's father abdicates. He wanted to marry Aoi but was denied the chance. He is engaged to the Sixth Princess, Kokiden's daughter. He begins his rule in Chapter 10 and is best described as ineffectual emperor who allows others to control him. He exiles Genji and against his mother's wishes brings him back. He abdicates the throne in Chapter 14 and has a son by Lady Shokyoden who is made a crown prince after Genji's son. During Genji's absence he suffers from a mysterious eye ailment. At the end he becomes a monk.
Aoi: Genji's wife. For a while the crown prince, Genji's brother, sought her hand, but Minister of the Left decided to wed her to Genji. She is somewhat older than Genji by four years and had unease with him and is best described as cold. She offends Lady of Rokujo Mansion and is visited by ghosts who try to possess her. She dies after giving birth to Yugiri.
To no Chujo: is described as having an amorous nature and is the son in law of Minister of the Right. He is Genji's companion and rival of sorts. He recounts a story of a woman he knew who was never jealous and had given him a child. He is father of Kashiwagi and has other countless children. Very often he tends to lose out to Genji. He is married to fourth daughter, Kokiden's sister.
Lady of the Locust shell and her brother: She is married to the governor of Iyo and her real brother became step-brother by marriage. Genji had fallen for her somewhat, but due to the different stations in life, she keeps refusing him and disliking him. Its possible that Genji also has an affair with the brother as well because of the woman's stubbornness. Later on the brother works as one of Genji's pages. Lady of the Locust has the robe returned to her and leaves with her husband. After she comes back, her husband passes away and his sons don't look after her, thus she decides to become a nun. Her brother is in Fifth Rank.
Lady of the Locust Shell's stepdaughter: She is described as lively and Genji neither likes nor hates her. He mistakes her for Lady of the Locust Shell. She is larger than the stepmother and isn't experienced when it comes to love. She is also described as lighthearted and inattentive and had hopes that Genji would write to her. She got married to a guards lieutenant.
Lady of the Evening Faces: She was possibly To no Chujo's lover who refused to be jealous and who bore a daughter to him. She met up with Genji and when they went to the mountains a malignant spirit, most likely Lady of Rokujo Mansion, has killed her during the night. She is described as an orphan and comes from an undistinguished background. She is an orphan and her father was a guardsman.
Lady of Rokujo Mansion: Genji's mistress who is considerably older than he. Genji managed to get her to like him but then dropped her. She is described as jealous, and is thought to be responsible for Aoi's and Lady of the Evening Faces deaths.She was married to Genji's uncle who died, other crown prince and had a daughter named Akikonomu.
Murasaki: When Genji discovers her in the mountains she is ten, suffering the loss of her mother and being raised by a monk's sister, possibly grandmother. She heavily resembles the unattainable Fujitsubo. She seems to like Genji and early on in life she claims him as his own. Genji takes her to make her into his ideal woman. She is Prince Hyobu's daughter, which makes her Fujitsubo's niece. Shortly after Aoi's death, Genji makes her his wife. She is childless and adopts Genji's daughter from Akashi lady. She also has helped raise Genji's grandsons and granddaughters, Third Prince being among her favorites. She has always wanted to be a nun, but Genji never allowed her to become one. She died in 40th chapter without fulfilling her desire.
The Hitachi Princess/Lady of the Safflower: She is talented at playing koto and was raised in old-fashioned dignity and lives in dilapidated house that is described as romantic. For a while both Genji and To no Chujo pursue her, but she is impossibly shy. She is also a daughter of Prince Hitachi. She is pretty, aside from the nose that Genji continues to obsess about. For New Years she sends him old fashioned clothes of red color.
Naishi: She is of advanced years and is described as wellborn, talented, cultivated and widely respected, although she wasn't discriminating in matters of heart. Genji and To no Chujo have affairs with her despite her brazen nature. She is also good at playing lute and is nearly sixty. If I"m not mistaken she becomes a nun, and Genji inherits the task of making sure she's okay.
Oborozukiyo: The Sixth Princess (Kokiden's younger sister) She meets Genji during the Festival of Cherry Blossoms during the end of the Second Month. Genji has an affair with her and they exchange fans. She continues to have affair with Genji, gets caught and due to her, Genji becomes exiled.
Lady of the Orange Blossoms: Her elder sister was the emperor's concubine and she is having an affair with Genji as well as completely dependent on Genji.
The Gosechi Dancer: Possibly a charming girl who is a daughter of the assistant viceroy of Kyushu. She enjoys being in Suma.
The Akashi Lady: Her father is a monk and is best described as extremely proud as well as intractable. He refuses to marry her to the incumbent governor and begins to scheme to have her become Genji's. She is described as someone who's not a great beauty but someone who is intelligent, sensitive and a gentle grace and is reconciled to her sad lot. She doesn't want to marry someone close to her station and if let behind she would either become a nun or throw herself into the sea. Later on she bore Genji his only daughter and has to give her up to Murasaki.
Reizei Emperor: Was born during the Second Month. Is thought to be the son of the Emperor, but in truth he is Genji's and Fujitsubo's son. He heavily resembles Genji and is heavily doted on by his grandfather. Described as radiant. He is adopted by Suzaki Emperor and around Chapter 14 becomes the emperor himself. Akikonomu is his consort, an empress.He only has a daughter.
Third Princess: The favored daughter of the Suzaki Emperor. When Suzaki Emperor is ready to become a monk, he worries about her the most and asks Genji to marry her as well as take care of her. She is best described as childish, naive and inexperienced. She also bears Kashiwagi's son and due to the life she becomes a nun. She doesn't seem to have any feelings towards Kashiwagi.
Yugiri: Son of Genji and Aoi. He was born early. He is described as beautiful and arouses forebodings. He is to have elaborate education, and around the eyes he bears resemblance to Genji's other son. He falls in love with To no Chujo's daughter named Kumoinakari and moons over her a lot. He also falls in love with Genji's servant's daughter as well, a Gosechi dancer. He is intelligent, and isn't fanciful, although he resents that he is forced by Genji to not be able to keep up his appearances. He has numerous children and is "monogamous". He is best friends with Kashiwagi.
Akashi Princess: Daughter of Genji and the lady from Akashi. In order for her to have a future, she is adopted by Murasaki. As a baby she had a sweet and docile temper. She gets involved with a crown prince (Suzaki Emperor's son,) and gives him multiple sons and daughter.
Kashiwagi: One of To no Chujo's sons who at first is in love with his sister Tamakazura without knowing it, and is also best friends with Yugiri. Later on he falls in love with Third Princess, Genji's wife and fathers a child on her by name of Kaoru. Due to him worrying about Genji finding out, he begins to decline and eventually dies. He is talented when it comes to music.
Kaoru: Son of Third Princess and the world thinks he is also Genji's child, although in reality, Kashiwagi fathered him. He is known as the "Fragrant Captain." He is very popular and much sought after by what seemed to be everyone. Reizei Emperor looks out for him. He is best described as very serious, has a unique fragrance, is also very cynical and isn't interested in amorous adventures, although he had a lot of them. He meets Eighth Prince and eventually his daughters Oigimi and Nakanokimi. He falls in love with Oigimi but she would prefer that he marry her sister Nakanokimi. After Oigimi's death, he tries to love Nakanokimi unsuccessfully followed by Ukifune. He is unlucky in love.
Niou: Son of Akashi Empress and Suzaku son who became Emperor after Reizei Emperor abdicated. He is known as "His Perfumed Highness." He competes with Kaoru in everything, the fragrance being the chief competition. He was Murasaki's favorite and loves women, although none seem to excite him to marry, aside from Reizei princess.
Oigimi: The elder daughter of the Eighth Prince and his wife. She is described as comely, has a gentle disposition, elegant in face and manner and has a suggestion behind the elegance of hidden depths. She is superior in quiet grace. She is good with the lute and both she and her sister live in Uji. She is loved by Kaoru and Niou eventually. After Kaoru displeases her in some manner, she starves herself to death.
Nakanokimi: Oigimi's younger sister. After her birth, her mother fell ill and passed away. She was taught thirteen-stringed koto. Niou falls in love with her and eventually she becomes the principal wife. She bears him a child. After the death of her sister and her father, Kaoru pays her unwanted attention, and wanting for him to leave her alone, she tells him about the third sister, Ukifune.
Ukifune: She is half sister to Nakanokimi and Oigimi. In beginning she was going to be married to someone, but when her suitor learned that she is not the favored or real daughter of her stepfather, her mother makes a choice in going with her to somewhere else. She ends up meeting both Kaoru and Niou, although she loves Niou a lot more, and eventually causes everyone to think she killed herself when in fact she decided to give up the world and became a nun.
The story is unfinished, but the author lived more than a thousand years ago.