In some way, Run considers Ren as her big brother. They care about each other very much, and one always helps and respects each other before they became two separate individuals.
In the beginning the two were always at odds with each other do to their respective feeling for Lala and Rito. With Run being in love with Rito and seeing Lala as both an enemy, do to being unintentional harassed as a child, and a love rival for Rito's affection, and Ren being in love with Lala and believing Rito to be unworthy of being her fiancé.
In Darkness, after the two have separated, Ren has since stoped his interference in Run's advancements towards Rito. The two seem to have a respectable relationship as siblings.
Because of a kiss Ren accidentally received from Rito, Run has been in love with Rito madly and initially, she will try every means to win Rito's heart, including using her body to seduce Rito, which usually ended up in total failure. However, recently, because of her work as an idol, she displays her presence less frequently but she still love Rito with a passion, and so far in the series, she is the one of the girls who ever confessed her feeling to Rito, although Rito turned her down, thinking that he did not deserve her feeling. However, that does not extinguish her feeling, as it just serves to make her burning love leap higher by saying that she won't lose to anyone (to win Rito's affection). Run has since expressed that she has no intention of sharing Rito with anyone else and that she wants him all to herself.
Initially, Run dislikes Kyouko because she is just a guess actor in Kyouko's TV show. However, charming by Kyouko's kindness, and learning of Kyouko's true identity as an alien, the two have since then become best friends, and Run would always seek Kyouko's help for her feeling for Rito. She does not know that Kyouko also developed some crush towards Rito as of yet.
While Run's male counterpart, Ren madly in love with Lala, Run hates Lala with a passion, not just because her childhood (with Ren) has been a test subject for Lala's bizarre inventions, but also, Lala is a love rival when it comes to Rito Yuuki, and she would like to play pranks on Lala, however, all her pranks ended up in failure and some even backfired on her.
Lately, because of her work, she did not have time to "take her revenge" on Lala anymore. Still, she considered Lala as her greatest rival, and she would like to beat Lala to Rito's heart.