Both of his parents are employed and unnamed. Taishi is the younger brother of Saki Kawasaki. He also has a younger sister, Keika, and a younger brother who is unnamed.
Saki is Taishi's oldest sister. They have very good brother-sister relationship. Being the two oldest children to overworking parents, they are very mature and both take care of their younger siblings. Taishi is often concerned with Saki's welfare and notices when her behavior changes for the worse. They care about each other a lot, evidenced by Taishi seeking help about his sisters late nights and declining grades. Saki also requests Hachiman's help for Taishi when he wants advice for getting into Sobu.
Throughout the series it is mentioned that Taishi and Saki talk a lot at home, they have a very similar relationship to Hachiman and Komachi have. Hachiman even ironically refers to Saki's behavior towards her brother as "Bro-con"..
Komachi introduces Taishi to the Service Club when he needs help with his sister and is concerned about her recent change in behavior. After resolving his first request, Taishi enthusiastically tries to help with any request he can.
Hachiman has an immediate distrust of Taishi because he is friendly with Komachi. When first meeting him, Hachiman described Taishi as "a boy with good prospects, but still not good enough for Komachi". Hachiman wrote Taishi's name in his "Definitely Unforgivable Person List", because of his closeness to Komachi.
Hachiman sometimes refers to Taishi as "that thing". Despite Hachiman's somewhat rude treatment, Taishi is rather fond of him and affectionately calls him "Onii-chan" (bro/brother), much to the latter's annoyance.
Taishi is impressed with Hachiman's vocabulary and dependability. He begins to respect him when Hachiman helps him solve his family issue.
Taishi has a large amount of respect for Hachiman and asks for his help or advice occasionally. Hachiman initially helped Taishi with a family dispute, although it was only because of a Service Club request. Later, he freely helps Taishi when asked by Saki or Komachi. He gives Taishi advice about choosing a high school and the requirements to enter Sobu High. He also discussed beautiful girls in Sobu High School when Taishi asks for his opinion, referring to Yukino Yukinoshita. Hachiman describes Class 2-J filled with 90% girls, and also advises him to give up, as the girl he liked doesn't necessarily need to like him back.
Another indication that Taishi likes Hachiman is whenever Saki and Hachiman talk, she sometimes mentions that Taishi talks about him at home, indicating his respect and fondness for Hachiman.
Despite their unusual and one sided relationship, Hachiman seems to understand Taishi's problems and thoughts better than Komachi or any other Service Club members, even better than Saki herself.
Taishi told Hachiman his impression of Yukino is beautiful yet frightening. He seems to be in awe with the unparalleled beauty of Yukino and is thankful to her for the Service Clubs help. He mentions to Hachiman he hopes to get girlfriend like her if he joined Sobu High school, although someone less scary.
He views her in the same light as Yukino, a beautiful and helpful high school girl. However nothing is explicitly mentioned. They have no one on one interactions.