If you have ever seen a girl across the room that you consider absolutely gorgeous and completely out of your league, don't dismiss your chances with her too quickly. Just because a woman is attractive does not mean she is only interested in speaking with other people that are extremely attractive as well. As you get to know her, you might find that even though you felt she is out of your league, she considers you to be in her league based on your personality and shared interests.
Set your expectations at a reasonable level. Don't expect to have her fall in love with you after your first conversation, or want to go on a date with you. Instead your goal when flirting with someone should be to at least hold their attention and start to get to know them.
Break the ice by joining in a conversation she is having or comment about something that you think you two might have in common based on something you heard her say.
Continue the conversation just like you normally would with a friend. Maintain eye contact. If you fail to maintain eye contact she could interpret it as a sign that you are trying to hide something from her. Also if she catches your eyes wandering to view other "assets" she might get offended as well.
Keep your confidence up. You may be one of many who find her beautiful, but you may be on of just a few who have the confidence to have a conversation with this girl. The door to flirtation opens with a conversation, as that gives you a chance to proceed to the next step.
Compliment her if the conversation is going well. Issuing a compliment at the start of the conversation might get you labeled "creepy" or she might wonder what your motivation is. If you have already developed a conversation though, it will come across as more natural and less scary.
Try to close the conversation with another opportunity to talk. If she really seems interested in you, try asking her out for a date. If you aren't sure yet, invite her to go to a group activity with you. This will help you spend more time and talk with her, without her feeling like you are trying to get her alone, and reduce the risk of her being scared off.
- Be natural, don't go in with an agenda. If you do, your conversation will feel forced.
- Do not be intimidated by her appearance. Remember, even attractive people have challenges, including finding that others are scared to talk to them out of fear of rejection.