People tend to want things they can't have and will go through several hoops to get it. In the case of relationships, the taken man is a prime example of look, but don't touch. If you think he is the one for you, then there are things you can do to better your chances and help sink his relationship with the other woman.
Create competition. If you want to spark the embers of breakup, you need to create a secondary object of attraction other than the girlfriend. If you want to be the new girlfriend, make yourself prettier than his current girlfriend. Men are visual animals and if you want to make a strong impression, then look sexy. He will definitely notice. Your looks can be your gateway into his circle.
Become his confidant. If you are going to break up the happy or unhappy couple, you need to know about their relationship. Listen to him and become a welcome ear when he is angry or frustrated. If he believes he can trust you, you can learn about the couple's problems. You can also offer helpful advice, including "Maybe she just isn't right for you."
Be discreet. If you want a relationship and not just a one-night stand, then be close--but do not do anything to be overtly sexual or to betray your true feelings. If he begins to think you are trying to sabotage his relationship, he may cut off ties with you. Be his friend, flirt a little, but keep your touching and feelings in check. Wait until the girlfriend is out of the picture before you start the heavy flirting.
Fan the flame of jealousy. Nothing ends a relationship faster than jealousy. You can make a point to be seen with him while she is around or even talk to him while he is with her. Let him introduce you to his girlfriend and the girlfriend will begin to see red. If this goes on long enough, she will mention you to him and he will say you are just friends. An argument will begin and a breakup may not be too far off.
Feign ignorance. If you can keep this going long enough, eventually the girlfriend will confront you. Men are naturally obtuse when it comes to women and he wouldn't have figured out that you were trying to break up the relationship, but the girlfriend will. She may even confront you with her boyfriend. It is best to play dumb and say you are only being a good friend and that you would never try to steal someone away from a girlfriend. The guy will likely believe you, but the girlfriend will see right through you and get madder and madder. This may be a deal breaker for him--especially if he thinks his girlfriend is going to go crazy every time he has a female friend.