Okita and Sexuality
First, I have to say that Okita is not a very sexual person. In fact, he's one of the few Shinsengumi members who I really have to go off complete speculation for, because he doesn't so much as make a dirty joke in the whole of the series, much less (like some of the others, say) visit the pleasure quarters or keep a mistress. I could interpret that as complete innocence if I wanted, but he's a twenty-three-year-old man whose best friends pretty much ALL go to the pleasure quarters/keep mistresses/et cetera. I therefore choose to interpret it as knowing what's up and just not really caring very much.
Furthermore, he doesn't like being touched and doesn't touch other people very often. That's not the behavior of a man who'd feel comfortable having casual sex. He gets pretty touchy about his physical boundaries with everyone except Kondou and Hijikata; even Tetsu, who is probably his closest friend outside of the Shinsengumi leadership, gets complete withdrawal and refusal to move or acknowledge the touch when he grabs Okita angrily by the collar. The people he willingly touches in the series are: Kondou, once, a glomp while trying to wig out the Ichimura brothers. Hijikata, several times - backrubs, sharing an umbrella, tugging him around by the hair, accepting support from him when hurt and sick. Tetsu, once, a hug when Tetsu was sobbing over a man having been killed a few inches from his face. The lesson here, clearly, is that Okita is only physically comfortable around people he trusts with more than his life. No one on the Island is in the list of people he'd touch casually, Saitou and Yukina probably being at the top as the only people he's touched period.
Now that everyone thinks Okita's a pure-minded virgin, let me go on to say that I seriously doubt, at his age and given his life, that this is the case. I suspect he had girls throwing themselves at him like crazy in his mid to late teens, being the attractive well-spoken young samurai that he is. And he has enough curiosity that he probably slept with one or two of them to see what the big deal was. For Okita, sleeping with a woman he doesn't know really really well would probably boil down to (1) he would top. Completely. Even if he didn't know what he was doing. Because he wouldn't want the other person to take control unless he trusted them a lot. This probably means that (2) it was pretty bad sex and he wasn't particularly impressed. As a result, he sort of filed the experience away under "slightly embarassing, not all that fun, I guess you people can go with it if you like that sort of thing." I doubt he's ever slept with a man, because given his looks the sort of people who would hit on him are the sort of people who would expect him to play the woman in bed, and, um, I suspect Okita would respond to that with something along the lines of "How about a sparring match first? ♥" and break something important of theirs. Like a rib or five.
Hijikata - who does too need his own paragraph in the sexuality essay, shh - I don't think has ever asked. At least, I've decided to play it as "regardless of the nature of Hijikata's feelings about Okita, they are not in fact lovers at the moment" ... and the only way I can reconcile that with Okita's behavior is that Hijikata hasn't asked and Okita doesn't think sex is important enough to ask. Because he is someone to whom Okita would willingly bottom, and although Okita would hesitate first because he's seen Hijikata woo, bed, and dump a lot of women I really doubt he would actually say no. I really doubt he actually knows how to say no to Hijikata. And I think that it'd probably be a good experience, along the "suddenly I get why people care about this" lines. But I don't think it's happened, and if it did, I think it was a teenage fling that didn't last too very long and that probably started with Okita being the one to seduce Hijikata. Again, things I'll have to work out with any potential future Hijikata apper, I can play it either way but I'd like consistency.
As for who Okita is likely to feel twinges of sexual interest towards, um. So far in-game, the only person who he's noticed that way in the slightest was Saitou with the whole "show me your oni side" thing. Risking his life in battle is a turn-on. Enough of one to make him actually notice Saitou that way outside battle, definitely not although Saitou's notable resemblance to Hijikata in looks and personality DOES register. Enough of one that he's rather pricklier about Saitou calling him a girl, well, yes. Physically, his type is, in men, tall and well-built, and in women, slender, not too curvy, not weak. Also, he has to like you personally a whole lot. Being able to beat him when he's fighting full-strength against you helps, too. Even pushing him really hard in a full-strength fight will make him aware of you in a way that's best described as "homicidal with an option of sexual." It's one of the few times in his life when he isn't holding anything back at all, which makes him a lot more likely to consciously register impulses like that.
The likelihood of his actually becoming romantically involved with someone in game is just about zero. If one of the people who he really likes decided to seriously pursue him, there's a chance, but I can't imagine it would be in character for any of them (the list right now is Yukina, Saitou, Wolfram, Senri doesn't count on the grounds of being his student, Naoto doesn't count at the moment but might, particularly if she gets better fast, Presea and Yuki don't register because he hasn't really talked to them enough to be further than the 'nice young girl' stage of recognition.)