Is the computer itself on and just the display isn't, or is the whole thing not coming on. If it is the the whole laptop not coming on, try this and see if it will start up.
1. First make sure the device has been charged.
2. Unplug your charger cord.
3. Remove the battery.
4. Press and hold the "Power" button for 30 seconds and see if the device will turn on.
5. If it does, great, now plug back in the power/charger cord. (If it didn't turn on without the cord in, it should turn on once you it is in. Yes sometimes they will turn on, or try to turn on, even with nothing attached/plugged in.)
6. Now put the battery back in.
If it isn't the whole laptop, but just the display, try turning off the laptop, connect an external monitor, turn on the external monitor and then turn on the laptop. Can you see things on the external monitor? If yes, then it is the display or its connections that are not working. If nothing shows on the external one either, then your graphics card probably needs replacing.