Green light on but battery stays red

My hunch is that the battery is in poor condition. I would recommend downloading coconutBattery and checking the percentage of the design capacity, the status, and the cycle count. If the percentage of the design capacity is less than 80% or the cycle count is greater than 1000 or the status says “Poor” then you may have a bad battery. Unfortunately these batteries are embedded and are somewhat difficult to replace. See this guide for more information.

If the battery isn’t failing the next things to try would be replacing the charger or replacing the MagSafe port.