Never remove a battery while the car is running. If the alternator wasn’t charging then that is reason to change it out, a battery while not maintain a correct charge without a properly functioning alternator.
The easiest way to determine if you have charging or battery problems is with a multimeter. Sitting in your driveway with your car not running your battery should read 12.5-12.7 volts. Less than 12.5 means your battery is getting older. 12.2 or less is essentially a dead battery.
12.2 will start your car during the summer, however the first time it gets below freezing your car won’t start or will barely start.
With the car running you should be able to measure between 13.8-14.2 volts at the battery terminals. Less than 13.5 and the alternator won’t keep up with the electrical demands of your car.
Of course sometimes connections are loose or corroded, which can also be measured using the ohm side of the multimeter.
YouTube is a valuable resource to learn how to use a multimeter and Walmart , Home Depot or any auto parts store is a good place to buy a cheap digital multimeter. Usually less than $40.