First, reboot computer to BIOS/UEFI. Check to see that the drive is seen here. If it isn't, double check your cabling. If still not seen, there's a possibility that the reason your laptop screen went out is because the drive went bad.
If drive is seen in BIOS/UEFI, use the boot priority to boot from it. If you get error messages about bootmgr or no OS being found, see if you can connect the drive using the SATA-to-USB converter/adapter that external drive enclosures use (you'll see them if you open up the external drive enclosure).