1) Remove battery, and try AC Adaptor. If it powers the laptop without problem, then I would expect the problem to be the battery, or the charger. If still a problem then Laptop, or charger.
2) while it can be done, it is difficault to determine which is which other than trying a KNOWN good charger.
3) To test you need a Digital volt meter (About $17 @ walmart), and the electrical ability to either make an adaptor cable, or ability to splice into the cable coming from the AC adaptor.
.. To measure voltage, you simply go across the two wires for 19 VDC.
.. To measure the current you have to cut a wire and put the meter in series.
If the Current is above the rated value - it's the laptop.
If Current is Below the rated value and the Voltage is low, it's the AC adaptor.
Note : If you don't know how to do please get help from a valid technician.