These are the relationships of the Fire Ninja Kai.
Nya is Kai's younger sister. She was the only family he had when their parents disappeared while they were young. Because of this, Kai is a little overprotective of his sister as he wanted to keep her safe from danger. It was this side of him that caused Kai to even become a ninja as he just wanted to protect her. Kai was also the first ninja to learn about Nya's identity of samurai x and agreed to keep her secret from the other Ninja.
In Season 5, when Nya begins training to become the Water Ninja, Kai was excited that his sister was the new addition to the team. He was very impressed to see Nya unlock her true potential and defeat The Preeminent leading to the ninja achieving victory over the ghostly threat.
In "Day of the Departed," he and Nya pay tribute to their parents and wonder if they learn what happen to them before the ghost of Chen attacked them. The two siblings fought against their enemy and showed great teamwork as they manage to defeat Chen once more.
During The Hands of Time, Kai informs Nya that their father was still alive and he was helping their enemies the Time Twins though Nya refused to believe such a thing until Kai proved it to her. When they come face to face with their father, Kai attacks him as Nya tries to stop him as he was still their father. After their mother appeared, she explained to Kai on what really caused them to serve the Time Twins. Kai and Nya were soon forced to help the Time Twins when they threaten their parents and a weaken Wu. After managing to obtain the last time blade, Kai and Nya pursued the villains as they time travelled to the past and after a long battle across Time they manage to emerge victorious at the cost of Wu. They return to their friends and save their father who they happily reunite with alongside their mother after many years apart.
Ray is Kai and Nya's father, was a blacksmith, and the one who Kai inherited his powers. He disappeared sometime after Nya was born and Kai often wondered about what happened to him and their mother. Kai took over his father's blacksmith shop and swore to surpass his father's skills, even though his talents were far behind his. In Season 4, he was interested in hearing the hidden story about his parents' disappearance from Chen, who promised the information if he betrayed his friends.
In "Day of the Departed," he and Nya pay tribute to their parents and wonders if they would find out what happened to them.
In The Hands of Time, Kai was shocked to hear from Acronix that he knew Kai's father and caused him to question everything. Later on, he is horrified to hear from Krux about his parents being traitors who aided him and his brother. Kai then begins to dive deep into his parents' past and discover that his father did in fact aid the Time Twins. Kai was angered by this revelation and when reuniting with his father, he labeled him a traitor and attacked him until Nya and his mother Maya stopped him. Kai learned that his parents were forced to help the Time Twins after Krux threatened to harm him and his sister when they were children which caused Kai to feel guilty over trying to hurt his parents and apologized for accusing them when they were really protecting him and his sister. Kai later saved his father from succumbing to the affects of the time blades and was overjoyed to have him back after so long.
Maya is Kai and Nya's mother, she disappeared sometime after Nya was born and Kai often wondered about what happened to her and their father. In Season 4, he was interested in hearing the hidden story about his parents' disappearance from Chen who promised the information if he betrayed his friends. In "Day of the Departed," he and Nya pay tribute to their parents and wonder if they would find out what happened to them.
In "The Hands of Time," Kai was shocked to hear from Krux about his parents being traitors who aided him and his brother. Kai then begins to dive deep into his parents' past and discover that he did in fact aid the Time Twins. Kai was angered by this revelation and when reuniting with his father, he labeled him a traitor and attacked him until Nya and their mother arrived and stopped him. Kai learned that his parents were forced to help the Time Twins after Krux threatened to harm them which caused Kai to feel guilty over trying to hurt his parents and apologize for accusing them when they were protecting him and his sister. After Kai managed to save his father from the affects of the time blades, he was overjoyed to have both his parents back after so long.
Kai first encountered her on the ferry to Chen's Island while there he defended her from Karlof. He developed in a romantic interest in her but was shocked to learn she uses his element thinking they may be related but soon learns her power copies other which brings back his feelings for her. During the Tournament, Kai grows to trust Skylor especially when she shows information regarding Cole and Zane. He is heartbroken to learn that she is Chen's daughter and was secretly working with her father to undermine his friends. His feelings for her seem reciprocated, as Skylor couldn't look him in the eye when Chen captured him. She also seemed touched when he mentioned that he fell for her. Skylor soon turned against her father and was captured as a result but Kai went off to rescue her from Chen.
Kai's feelings were so great that he accepted her new Anacondrai form which allowed him to unlock his Elemental Fire Dragon. When she was sad about people running from her new appearance, he cheered her up by mentioning she has changed by joining his friends. The two fought together to face Chen and his army emerging victorious from the conflict. After the battle, Kai offers her a spot on the team but she declines to take over her father's noodle business. She tries to kiss him on the cheek but Dareth interrupts them but she promises to keep in touch.
Kai is Master Wu's student, along with the other Ninja. In the beginning of the series, Kai was recruited by Master Wu to become a Ninja. At first, Kai didn't care about training with Wu even considering him a crazy old man. But as the series went on, they grew closer. Kai and the other Ninja cared about their master and is glad to have him mentoring them.
In Season 7, he and the other ninja come to his aid when he fights Acronix by himself when Nya saves Wu from succumbing to the effects of the time blades. with Wu back to normal, he helps them foil the time twins plan. He later aids Kai and Nya battle the Time Twins but when the villains prepare to finish the ninja off, Wu sacrifices himself to save them. Though happy at their victory, Kai couldn't help but be worried when his master was feared lost in time with their enemies. He and the Ninja elect Lloyd as their master and are happy to hear his first order is to discover what happened to Wu.
Initially, the Ninja and Lloyd had a bitter relationship due to the fact that they prevented him from ruling Ninjago. When he redeemed, Kai realized his destiny was to protect him. Though Kai was jealous that Lloyd was the Green Ninja instead of him, that doesn't hinder their friendship.
In Season 5, after Lloyd was possessed by Morro, Kai was the most determined out of all the Ninja to get their friend back. When Morro left Lloyd's body and threatened him, Kai gave up the Realm Crystal and leapt into the water (despite his fear and inability to swim) to save Lloyd from drowning. He was relieved to have his friend back and comforted Lloyd when he was upset about dropping his guard by telling they would defeat any threat. Kai's words proved true as the Ninja were able to save the city of Stiix, defeat The Preeminent, and her army of ghosts.
In Season 7, he and the other Ninja come to the decision to make Lloyd their master after Wu is lost in time with Time Twins. Kai most likely considers Lloyd as a close friend or even best friend following Season 5.
They are good friends throughout the series. In the earlier seasons he and Cole were the leaders of the team and sometimes they rivaled over the leadership.
In Season 5, Kai's and Cole's friendship grew stronger in this season. Kai very sad that Cole became a ghost and attempted to move him out of his sadness. Throughout the rest of the season, he motivated Cole to never give up and to stay by his side. Overall, Kai respects Cole and they are close friends. Both would put the other Ninja before themselves.
In "Day of the Departed," Kai was very happy to see Cole return to being a human.
In Season 7 Cole and Kai didn't interact much. Although, Cole was very worried about Kai after he and Nya follow the Time Twins through time, along with Wu.
Like all the other Ninja, he initially thought Zane was weird, but when he found out that Zane was a Nindroid, he and the others supported and accepted their friend. He and Zane soon grow closer throughout the series. When Zane sacrificed himself to defeat The Overlord was seemingly destroyed, Kai was greatly upset by his friend's demise that he couldn't be a ninja anymore.
In Season 4, Kai was sad at his death, being upset that Zane died instead of him, out of guilt. When Lloyd suggested the idea of replacing Zane, Kai and the others protested at this with Kai even stating that without Zane maybe there was no team. When Kai and the others learned that Zane was still alive, they were very surprised at this and entered the Tournament of Elements to locate him. Kai was glad that Zane was okay and reunited with his fellow Ninja. In Season 5 Kai and Zane rarely interacted, but Kai wasn't ready to follow Zane's leadership. He laughed at Zane's malfunction voice most of the time. Zane respects Kai as a friend and Ninja and usually follows Kai's leadership. Overall, Zane and Kai get along great.
Jay and Kai are good friends throughout the series. They didn't talk much in the beginning, and Kai didn't like it when Jay starting dating his sister, but he soon accepted this. The two have grown closer throughout out the series. On missions, Kai and Jay will team up and make a great team. Kai sometimes gets annoyed with his childish behavior, but they both like to tease Cole and Zane.