To troubleshoot a laptop screen that doesn't turn on, first listen closely to the laptop to determine whether it has actually booted. If it has not booted properly, try connecting the power cable if the laptop has been running on battery power. If the computer is connected to a docking station, disconnect it. Peripheral cables such as USB devices or an external mouse should also be disconnected.
It's also possible that the battery needs to be recycled or replaced. Try removing the battery and letting it sit outside the laptop for at least 1 minute. Replace the battery, and reboot the laptop to see if it is now powering the screen up.
If the screen is blank and the computer has booted properly, try shutting the laptop down completely and restarting it to see if the problem gets resolved during the reboot. If this doesn't work, check the brightness settings. Turn the brightness up to full power, and plug the laptop into an outlet to ensure that the computer is not in a power-save mode. Another option is to check the video adaptor. Plug a monitor or another type of display screen into the laptop. If that screen display works properly, then the issue is with the laptop display and likely requires the expertise of a computer repair person or store.