the power adapter supplies a fixed voltage as marked on the box. You can measure that with a DVM. Measure both DC & AC for excessive ripple. However, open circuit you should have 0V AC.
When plugged in with the battery inserted, there should be a charging light on.
The charging circuit is internal to the computer and some in the battery.
If the computer does not come on with battery removed and a good adapter plugged in, then it is internal to the computer. Could be a CPU or many other things.