This computer could definitely use a battery change. However, I would first try resetting the SMC by powering off the MacBook and plugging it in, once the charger light comes on hold down the left "SHIFT + CONTROL + OPTION + POWER" keys for around 5-10 seconds. Check if your MacBook will work normally without the charger now. If not, then I would recommend checking if your MacBook detects the battery by going to launchpad, then the "other" folder, then into "System Information". On the table that says "Hardware" select power and you will see information on your battery. You should first check the cycle count which is the number of times your battery has been charged. I recommended replacing the battery after around 800+ cycles. This is because the more cycles are on your battery, the more unhealthy it will become and the less charge it will hold. Also check if it says battery installed, if there is no battery detected then you need to replace the battery.