My older sister tried this on me a while ago, and it worked quite well. Everyone is different so you'll have to tweak and change things to suit yourself and her, but this is what I can think of.
Don't start of really pushy. Ask questions about school and what movies or books she likes. Once you've gotten conversations going, introduce things about your life and how you feel. Then go places and see films and eat meals together. Eating together is a great way to develop relationship, and to get to know someone.
Overall take it slowly. Ask questions and answer any you encounter. Have fun with it too. Being a sister is supposed to be fun, not just worry! I hope you two are able to get the ball rolling :)