How do I get a guy to like me and want to go out with me when he likes someone else?

First of all, if he is already in a relationship, personally, I'd stay clear of him if your only intentions are to "be" with him. Others might have different view on these, but I'm not particularly fond of "home wreckers" or people who try to develop a romantic relationship with someone who is already in one.

With that being said, it's really quite simple and obvious! Just hang out! If you two have a history or are just friends; just hang out like you already have been. As time goes on, try to see him more and more often, but remember he needs his space just as much as you need yours. Start with activities friends would normally do, then if you begin to start feeling that "vibe" take it to the next level and do more romantic things. You're not really "winning" him over, but more so developing a relationship from a solid friendship. Recently, I've seen more and more relationships develop without any romantic intentions from the get go. Take slow and eventually, it will progress into a relationship.

Others will recommend you try a more forward approach and flirt with him constantly, but if he already likes someone that can rub him the wrong way.

If he likes someone else, then I'd leave that be and accept the way things are... unless you would feel find possibly ruining somebody else's luck in life?

However if he is not seeing somebody else, and he doesn't like you for who you are, then I doubt there is much use in trying to win him for you... - unless he hardly knows you of course, in which case you might want to work on that...

Not the answer you're hoping for perhaps, but I hope it'll prove useful anyway. :)