It's hard to say if he likes her, the way we could say is romantic, but with the chapters focusing on them, he does show that he might be having mixed feelings. He wants to control her, as he claims. He wants to be the only thing on her mind. He wants to consume her life. But, does he like her?
I can't really say for sure, but I can say that he finds her alluring, in his own way. Yeah, his type is mostly older women with big boobs, or just big boobs in general, but at the same time, he also can't help himself around her. As some people pointed out, he doesn't use his knowledge about Kyoko to help her life, but I also feel like that isn't always the case. He remembers what she likes to eat for breakfast. He knows her interests and remembers the things about her that could be absolutely trivial under normal circumstances. Why remember the name she gave a cosmetic? Honestly, with how little he's supposed to care about her, why remember something trivial like that?
They have a complex relationship. He talks about her as his enemy, but who goes over to his enemy's place to check up on her after watching her mother say on national television that she doesn't have a daughter? Who sees her dead eyes and kisses her just to get a reaction out of her and snap her back to reality? To want to find her because he knows (now) that she tends to run away to cry somewhere? If he didn't care about her, why put forth the effort?
The Vie Ghoul (Beagle) incident was directly affecting him, so of course he'd have more at stake in it than other areas. But, he also did go out of his way to rescue her, climbing through a window, going through the forest, just to find her. To beat up a stalker to prevent him from speaking about anything he did to her to cover up everything? He didn't have to go that far to protect her. He didn't have to use underhanded methods to protect her.
He went out of his way to ask Mimori to find her in school and trap her so he could "kidnap" her to scold her about going to a fancy party and get dolled up by a man. Why would you do that if you didn't care? Even if you did care, why go through that much effort?
It's hard to say he likes her, but he does care for her. To say he doesn't would feel like a complete lie. He cares for her, but he's blinded by his ego to treat her as anything other than an enemy in her life. That's the only way he can interact with her.
I will say, I am deadly curious what they were talking about when he went to pick up the pickled gourd from the shop. The one where Ren lost his cool, entirely. They were laughing and joking with each other and she was smiling, so it makes me very curious what the deal was.
I hope we can find out about that morning when he left to America, because I can only imagine assume it was trivial on their part, based on the way Kyoko reacted when she felt like she covered her bases, but she did say he had a kind side to him. Even if it was out of anger to make Ren hurt for what he said to her.