Is it run on a lithium-ion battery ?
What kind of PC is it?
If its run on a lithiom-ion battery then read below:
Avoid frequent full discharges because this puts additional strain on the battery. Several partial discharges with frequent recharges are better for lithium-ion than one deep one. Recharging a partially charged lithium-ion does not cause harm because there is no memory. (In this respect, lithium-ion differs from nickel-based batteries.) Short battery life in a laptop is mainly cause by heat rather than charge / discharge patterns.
Batteries with fuel gauge (laptops) should be calibrated by applying a deliberate full discharge once every 30 charges. Running the pack down in the equipment does this. If ignored, the fuel gauge will become increasingly less accurate and in some cases cut off the device prematurely.
Keep the lithium-ion battery cool. Avoid a hot car. For prolonged storage, keep the battery at a 40% charge level.
Consider removing the battery from a laptop when running on fixed power. (Some laptop manufacturers are concerned about dust and moisture accumulating inside the battery casing.)
Avoid purchasing spare lithium-ion batteries for later use. Observe manufacturing dates. Do not buy old stock, even if sold at clearance prices.
If you have a spare lithium-ion battery, use one to the fullest and keep the other cool by placing it in the refrigerator. Do not freeze the battery. For best results, store the battery at 40% state-of-charge.
What model laptop do you have?
There is a much common problem with rechagable computers, as well as anything that needs a wire to do so. And this is precisely the most common problem the cable, If it does not charge at all. Like no LED is on while charging this is because of a conducting problem not battery, so the battery may be ok, but a more simple and cheap to fix cable could be the cause.
Have you checked the system bios?
It seems that there is something set in either the OS or the bios that might be set to stop the charging process while the laptop is on.
Another reason could be that where ever you are, the power is not enough to run your laptop and charge it?
I hope this helps.
I just sent it in to be fixed. It seemed that the power plug was broken.