Instructions
Go to your camp, marked on your in-game map, and speak with Leliana.
Go through the conversation, which will vary greatly depending on whether you are male or female. However, the general direction is the same; pick options that encourage more dialog and compliment Leliana. For example, the male warden can tell Leliana she is beautiful and charming while the female warden can ask her if she enjoys the company of women.
Return to the camp after completing several missions and speak with Leliana again.
Comfort Leliana, picking options that assure her that she can be comfortable with you and unafraid. The conversation will naturally lead to romance.
Ask Leliana if she still likes you or select the option to kiss her to initiate the romance scene.