Middle School
Kaori in middle school with friends/classmates
Kaori seemed to be good friends with almost all the girls in middle school and quite popular as well. This can be seen in Hachiman's middle school flashbacks. Kaori is often seen with other people or gossiping with her friends. The girls in her class felt sorry for Kaori because Hachiman asked her out.
Participating in Pre-valentine event
Kaori remains a popular and sociable person after middle school, as she got along fine with all the student council members of Kahin Sougou and Soubu High.
Kaihin Sogou High
Chika Nakamachi
Chika and kaori running for their double date with Hayato and Hachiman
They seem to be good friends and address each other by their first names. They are shown to hang out together after school as they ran into Hachiman at a diner. They also went on a double date with Hayato and Hachiman. In the light novel, Hachiman felt there may be a dent in their relationship after the double date. It is likely Chika felt Kaori was responsible for the double date ending poorly. Chika is not seen with Kaori after the double date.
Tamanawa
Tamanawa is the president of Kaihin Sougu's student council, which Kaori volunteers for. There are hints of Tamanwa harboring feelings for Kaori, but Kaori seems oblivious to them.
Soubu High School
Hachiman Hikigaya
Kaori and Hachiman seated in theatre on a double date
Hachiman explains his relationship with Kaori as just middle school classmates. She is a girl whom he harbored feelings for and he even confessed to her. Kaori rejected him and said it was a shock as they had never really spoken before. She described Hachiman as quiet and strange. Hachiman remembers his texting with her as it meant a lot for him to interact with a girl. Kaori however, remembers nothing about their conversations. They never spoke again after the rejection.
Her rejection was what molded him into the current antisocial loner that he is in the series. With this, Hachiman has even tried to avoid her (and all his other classmates) after graduating from middle school. Her rejection also caused him to start doubting the so called "nice girls"(like Yui) which greatly causing him to hold resentment and distancing himeslf from them.
It is debatable whether she still has Hachiman's contact info or not. In Light novel, it was hinted that Kaori knows where Hachiman lives but didn't see him in that region for a long time.
In the series, coincidentally, Hachiman meets her at a local café with a friend. Kaori recognizes Hachiman and even initiates the conversation showing no hard feelings or remorse about their past.
Kaori is still the same careless yet cheerful person she was in middle school taking Hachiman for granted until Hayato points out their flaws on the double date. After that she starts to see Hachiman in a better light, she started to wonders if he was dating either Yui Yuigahama or Yukino Yukinoshita.
Kaori's change in attitude towards Hachiman.
Later in the series, Kaori can be seen making attempts to reconcile with him. This was seen when she was nervous when meeting him and Iroha Isshiki and not saying much about their relationship but denied that they were close. Hachiman was very confused by her sudden change in demeanor and wondered what brought this change about.
Kaori offering Hachiman a canned tea drink
It likely Hayato's words she has come to see that there is more to Hachiman than she thought as she is surprised by Hachiman's new social circle and suspects Hayato to be his friend and the other girls (Yukino, Yui and Iroha) to be his love interest or dating relationship but Hachiman denied it. Likewise, Hachiman starts to see what kind of person Kaori is, and belittles himself at how he was in his younger years to have had a crush on her.
Kaori offering Hachiman friendship
During this meeting, Kaori offers Hachiman a canned tea drink and states that he (Hachiman) has changed to a greater degree or rather accepts that it was her fault for not seeing things clearly in the past for quickly jumping to conclusion without any thoughts. She soon brings up his past confession and says that Hachiman still isn't boyfriend material (in her opinion) though he voices he is no longer interested in her, but as it stands, they could be friends. Even though her answer to Hachiman's past confession remains the same, her tone and attitude is completely different and it gives off a more positive vibe. Kaori also invited Hachiman to their Junior high reunion to which Hachiman bluntly turned down which caused her to laugh thinking it was a joke. It is unknown if Hachiman ever considered her offer or whether he still intends to keep his distance from her.
Kaori borrowing chocolate mound from Hachiman
During the pre-Valentine's day event, they met again where Kaori asked Hachiman for his help and soon inquired if she ever gave him any valentines chocolate, which surprised him and earned the attention and jealously of Yui Yuigahama, Yukino Yukinoshita, Iroha Isshiki and Saki Kawasaki. When she is told that she didn't, she promised to give some to Hachiman now earning her his shock and the quiet ire of the girls. She went through with her promise as she did in fact give him a plate.
Hayato Hayama
Chika, Hayato, Kaori in theatre
Kaori is interested in knowing Hayama whom she feels has cool personality and as a pretext for her friend Chika who was also interested in him. She was introduced to him by Haruno. Kaori and her friend Chika exchanged contacts with Hayama. It was revealed that Kaori and Hayama were texting each other and ended up making plans to go out. Throughout the date, Kaori tries to appeal Hayama along with Chika, but they ended up on bad terms by the end of the date.
Hayama was disappointed with both of the girls and their behavior, he even told them off harshly on their double date.and an awkwardness was born between them,
During the Christmas Collaboration event she explain to Hachiman, how difficult it would be for her to face Hayama again.During the pre-Valentines Day event, she didn't interact with Hayato, despite being in the same place.
Iroha Isshiki
Kaori running into Iroha and Hachiman on their way to seminar hall
Kaori made her acquaintance with Iroha at the Christmas collaboration planning committee. Even though they run into each other on the double date incident, neither of them seems to recognize each other. Kaori suspected that Hachiman was only helping the planning committee to "score" with Iroha, which Hachiman completely declines.
Iroha was little bit cold after learning about Kaori's acquaintance with Hayato but failed to connect the dots to double date incident and also because of the change of topic related to Kaori's relationship with Hachiman.
Iroha recognizes Kaori as Hachiman's past acquaintance and was surprised Hachiman was close to someone in the past. Kaori nervously explains their past vaguely, leaving Iroha very curious about their past relationship.
Others
Haruno Yukinoshita
Haruno and Kaori met each other at a cafe as a mutual acquaintance of Hachiman. When Haruno learned of Hachiman's past and his confession to Kaori, she also noticed Kaori and her friend's interest in Hayato. Seeing that things were getting more interesting, she introduced them to Hayato. Haruno also followed them stealthily on their double date.
During the pre-Valentines Day event, Haruno and Kaori didn't interact. It is likely that Haruno is no longer interested in her while Kaori was either busy with her work or tried to avoid interaction with Hayato and his acquaintance.
Yukino and Yui
No interaction is shown between them other than the awkward and brief meeting on the double date event. Kaori initially has the wrong idea that one of them is Hachiman's girlfriend. Hachiman later clarifies that they are clubmates, but Kaori seems interested to know more about them.
It is clear that she doesn't know their names as she constantly refers to them as "those girls" or "the girls whom Hikigaya is close to".
Hayato made a rude comparison between them and her, stating that they are far more lovelier than she is.
During the Christmas collaboration event, they seem to recognize each-other. Kaori also praises Yukino and Hachiman for stirring things up in the planing committee, but referred to Yukino as "that girl".
Both Yukino and Yui seem unaware of Hachiman's confession to Kaori. Even though Hachiman mentioned his past confession to his classmate, they are likely unaware that the girl is Kaori.