The key to troubleshooting a laptop charger is to figure out if the laptop power problem is with the circuit board, battery, or laptop charger. The symptoms will all be the same---the laptop won't work. The most common failure point in this area is the battery. All batteries have a finite number of times they can be charged and will ultimately fail to hold a charge. The least common failure point will be the circuit board.
Charge the laptop by plugging the laptop charger into the electricity supply and the laptop. Try to charge the computer. If the computer won't charge, there's a problem with the system board, the battery or the charger.
Take a look at the power light on the laptop charger and see if there's a light on it. If there's no light, check that there is power supplied to the wall plug you are using. Try another device in that wall plug. If the other device won't work, the problem is with the wall outlet. Check the main electrical panel for blown fuses or tripped circuit breakers. Replace any blown fuses or reset circuit breakers. Test again with multiple devices. If the devices work, but the charger light won't illuminate, the charger is bad, replace it.
Connect the laptop charger to the wall and the laptop again. Check that the power charging light is illuminated on the laptop, and the power light is on the laptop charger. Try to charge the laptop again. If the laptop won't charge, the problem is with the battery---it's reached its end of life---or the circuit board is blown.
Remove the battery and plug the charger into the wall and laptop. Verify that the charger illuminates and the laptop illuminates. Turn the laptop on. If the laptop powers up, the charger and circuit board are fine, and the problem is with the battery. Replace it. If the laptop won't power up, the problem is likely with the circuit board. Service the laptop.