Try removing the battery, leave the power cord in, and then see if it will turn on. If it does, then your cord may need replacing. I say that because if a cord is not allowing enough power to get through so that it can both charge the battery and run the laptop, then it may well not even turn on when the battery is in. However it will when the battery is removed. I would also tend to charge only when turned off for the same reason, lack of needed power.
If this doesn't effect it, then you may need a new battery.