Why my computer fails to boot up

If it is plugged in and doesn't boot, there is an internal problem - assuming a good adapter.

If it boots when plugged in, and not when on battery, it is the battery.

Since the charge light does not light, that might indicate a bad adapter. That would obviously not charge the battery making the battery look "weak".

HOWEVER, the problem could be internal to the laptop.

1st - verify the adapter is good. Measured voltage at the connector should be slightly higher that what the label indicates. This does not mean it is absolutely good as it might not work under load. Testing under load is a little more difficult. The easiest thing is substitution - either using another known good adapter or another computer.

If the adapter is good, the issue is internal to the computer, since you have no charging LED lit. That could be as simple as a loose connector or bad solder connection - to as bad as a motherboard problem.