These are the relationships of the Fire Ninja Kai.
Family
Nya
Kai and Nya
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 decide to become a ninja so he could 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 will ever 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 was rapidly aging due to a blew by a Time Blade. They happily reunite with their parents after many years of being apart.
Ray
Kai and Nya reunited with their father, Ray.
Ray is Kai and Nya's father. He was a blacksmith, and the one who Kai inherited his powers. He disappeared sometime after Nya was born and Kai often wondered 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 ever 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 Nya 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 Nya. 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
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 ever find out what happened to them.
Kai and Nya reunited with their mother.
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 his parents 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 Maya arrived and stopped him.
Kai learned the truth that his parents were really 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 sincerely apologizes for ever accusing them when they were only 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.
Love Interest
Skylor
Kai and Skylor.
Kai first encountered her on the ferry to Chen's Island and while there he defended her from Karlof. From the moment they met, Kai immediately fell in love with her and developed a romantic interest in Skylor but was shocked to learn she uses his element, thinking they may be related but soon learns her power is actually Amber, 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 shocked to learn that she is Chen's daughter and was secretly working with her father to undermine his friends, but he also believes there is a conscience in her. 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 became convinced by Kai and 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 and decides to take over her father's noodle business, promising to fully redeem herself. She tries to kiss him on the cheek, but Dareth interrupts them. However, she promises to keep in touch.
Near the end of Season 6, Kai meets Skylor again (she was recruited by Jay to rescue the other Ninja from Nadakhan's magical sword) after he is freed from the sword. Kai thanks Skylor, saying that he is unsure how he can repay her. Skylor just replies that Kai "owes her one."
In Season 7, Kai goes to her for advice and she relates to his situation and tells him their parents left behind legacies, but they don’t define who they are; they are who they choose to be.
Friends
Master Wu
Kai and Wu
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. When Wu is back to normal, he helps them foil the Time Twins’ plan. He later aids Kai and Nya in the battle against 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 to be 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.
Lloyd
Kai and Lloyd
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 4, when Lloyd's father was banished to the Cursed Realm to defeat Chen, Kai made a promise to look after Lloyd when Lloyd was sad about his father's sacrifice and demise.
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 him that they would defeat any threat. Kai's words proved to be 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 the Time Twins. Kai most likely considers Lloyd as a close friend or even best friend following Season 5.
Cole
Kai and Cole
Kai and Cole are loyal buddies on great terms. They rarely bicker and get along very well because they share several common things: their confidence, courage, etc. 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 is 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 the latter and Nya followed the Time Twins through time, along with Wu. He is worried about what happened to Master Wu, and is determined to find him, no matter what.
Zane
Kai and Zane
Like all the other Ninja, Kai initially thought Zane was weird, but when he found out that Zane was a Nindroid, he and the others supported and accepted Zane as their friend. He and Zane grow closer throughout the series. When Zane sacrificed himself to defeat the Overlord and was thought to be destroyed, Kai was greatly upset over his friend's demise that he couldn't be a ninja anymore.
In Season 4, Kai was very saddened at Zane’s 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
Kai and Jay
Jay and Kai are loyal friends on good terms, though they interact only occasionally throughout. Nonetheless, they do truly care about each other. At first, their status was generally tense and turbulent since they were fairly distant through Season 1, rarely interacting. Their status became even more strained after Kai's sister became enamored with Jay, much to Kai's disgruntled chagrin. However, he soon respected and approved of their relationship.
As of Season 2, Kai and Jay have interacted more and become more agreeable. There are moments where Kai saves Jay from peril and comforts Jay when he is feeling down.