How do I recover a MacBook Pro with a nonfunctioning screen?
You could either get the screen repaired or connect it to an external monitor.
Repairing it could cost a lot, but that would be the best solution as your Mac would be fully functional again.  Take it to an apple store or call apple and they will know what to do.
If you are simply looking to get your data off of it or need a temporary solution, I would highly recommend connecting it to an external monitor.  That would be the easiest way to see if the computer is fried, or if it's just the screen.  Based on what you told me, it seems like your computer is fine just the screen is broken.