Need to Back Up a MacBook that Won't Turn On
Can you restart into Recovery HD? If you can, then you should be able to copy the HD onto an external HD using Disk Utility.

If you can't do that, but have another Mac available, you can restart into Target Disk Mode, and connect the two Macs together using a Thunderbolt cable. On the other Mac, your HD will show up as an external HD, and you'll be able to back it up.