Urie wants to use her so she could reveal information about the forbidden grimoire. He refers to her as his special butterfly. He claims that she is his and that he will hold her until she breaks. He is hinted to have feelings for her in episode 8, as his interest in her only increased when he believed that she doesn't have the grimoire. He also gets jealous when she and Rem dance.
In episode 9, he is shown throughout the episode looking for Ritsuka once he learns that she is the grimoire. He fights the exorcists, along with Mage and Shiki, when they threaten to kill her. In episode 10, all three confess that they love her, with Urie saying that he won't let anyone have his butterfly, calling her his pet. In the same episode, he is shown to be protective of her when he tells Rem that until he admits his true feelings they can't trust him with Ritsuka. This indicates that Urie no longer cares for the grimoire, and only wants to keep Ritsuka safe. This is confirmed in episode 11, when he says that he "won't let her go," and that "he'll cherish her always, even if she breaks." She is touched by his confession. In episode 12, he fights to protect her, and while he was saddened by her and Rem's love confession in the end, he was willing to let it happen since it was real. However, before he leaves he swears to return for Ritsuka one day. This indicates that his love for her remains an eternal thing in his heart, and that she is the only girl for him.
He and Rem are childhood friends. As of now they have a respectful albeit cold relationship, with Rem being the undisputed President. They are prone to frequent disagreements as shown when Rem prevents Urie from messing with his chess pieces, telling him not to try anything in regards to his plans and affairs, and interfering when he tries to hypnotize Ritsuka into taking off her protective pendant. Urie recognizes his friend's growing interest in and attachment to Ritsuka, even if Rem doesn't notice himself. Urie also has no problem with mocking Rem about how close he's been getting with her in episode five, bluntly stating that he's putting a lot of effort into what is supposedly just his "pawn."
In episode 8, when Rem and Ritsuka dance together, his eyes widen in surprise, hinting that he can tell Rem has fallen in love with her. In episode 10, he demands that Rem confess his love for her too, saying that until he does, and stops lying, none of them will trust him to keep Ritsuka safe. He is later satisfied with his confession in episode 11, and teams up with Rem to save Ritsuka, saying that he likes him better "straightforward." In episode 12, he was surprised by Rem's open display of love for Ritsuka, and because of it was willing to let them be together if they chose.
Urie and Mage have something of a rivalry going on. Shown in Epsiode 2, when they literally go head-to-head over their respective quirks. (i.e. Mage's manliness and Urie's flirtatious nature) This is shown again in episode 8 when they both ask Ritsuka to be their partner at the dance, they literally butt heads over it. He also gets jealous when Mage declares that he loves Ritsuka.
However, Mage does agree with Urie on occasion, and is not above working with him when he must. This is shown in episode 10, when he agrees when Urie says that until Rem can admit his true feelings for Ritsuka, then they can't trust him not to hurt her. This indicates enough respect to band together when a common threat is present. This is confirmed in episode 11.
Despite their rivalry, Urie tends to have to keep Mage in line at times. He has to remind Mage of the exorcists society as a place Ritsuka might be in and stop Mage from going ahead when Mage himself has no idea where Nesta's castle is located.
Urie doesn't like or respect Shiki, because he is a fallen angel, which he never fails to mock him about. He considers him beneath him, due to his fallen angel status, and dislikes it whenever Shiki tries to get ahead of him. More than once he has threatened to strike Shiki with his "thorny whip," and actually slapped him with his rose once. He gets jelaous whenever Shiki requests that Ritsuka be given to him, because he loves her.
After the song "Emo Liar," Shiki is shown agreeing with Urie that they can't trust Rem until he expresses his true feelings for Ritsuka. The two team up with the others to save Ritsuka.