Need more information before any help can be offered. What's the make and model of your Laptop and how old is it? What OS are you running? How old is your battery? Can you access the BIOS settings window? Have you tried connecting a monitor to your laptop and see if the monitor works?
You say the power light would blink, does it continue to blink or does it stay on?
You said the screen stays black, try this: turn all the lights off and start your laptop, then look very carefully at the screen. Is it totally black or can you see a very faint image in the background.
It's really hard to say until you provide more information, but it sounds like you either have a bad backlight or a bad inverter. If, when looking at the screen in a dark room with all the light off you can see very faint images in the background, odds are your inverter is bad. If the screen is totally black and nothing visible odds are its a bad backlight.
Again, without more information I just making a guess. Once you provide more information and I have something to work with I can help you resolve your problem.
I can tell you this for sure, if you hook up a monitor and it works with your laptop your problem is either the inverter or backlight. Either problem is easy to fix once we know for sure it's one or the other.
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