MacBook screen doesn't turn on
Try resetting the PRAM and SMC. 

How to reset PRAM: restart or power up your Mac and immediately hold down the Command ⌘+Option ⌥+P+R keys all at the same time. Keep holding them down until the Mac resets. Also, on these old models you used to have to do it a few times: i.e., just keep holding down and let it reset a few times, before it would be completely clean.

Resetting the SMC is a bit more model-specific and I can't remember off the top of my head what you needed to do with your model, but you'll find instructions here, or you can Google your model number with "reset SMC".