Technically all batteries in laptops are removable. How easy that is to do is a whole other story.
I am assuming you are trying to charge it while it is turned on. Try turning it off (not sleep or hibernate but off) and see if it will charge then.
If not, then you are going to have to determine if it is the charger cord or the battery that is the problem. You can try and borrow a charger to a similar device and see if it works or not. If a different one works on your laptop, then the charger is the problem. If a different charger doesn't work, then it is probably the battery.