The charging led will come on when the battery is charging. If this happens, chances are the power supply connector is working. If it flashes when you move the cable, the cord or the actual jack on the laptop may have issues(this happens.).
A notebook will shut down if it gets too hot, Please ensure the fan is spinning and if the laptop is older, you may have to look into cleaning or getting the heatsink cleaned.
While you may not see it from the outside, sometimes heatsinks get very clogged.
If the unit is running even for a short time you may be able to use 3rd party software like HWmonitor by cpuid to check the temperatures.
http/www.cpuid.com/softwares/hwmonitor.html