Car battery chargers come in different varieties for use in different circumstances. Some offer a full voltage charge to quickly recharge a dead battery in your vehicle. Others trickle charge a battery over a longer period of time. If you use a charger on your battery and the vehicle still won't start, there are a host of problems that could be causing the malfunction -- the battery may be dead and unable to hold a charge or the car's charging system could be fouled. The charger itself could be bad, which is a good place to start. You will need a voltmeter, which is a relatively simple and cheap device with a positive and negative lead, attached to either probes or alligator clamps. Voltmeters range from extremely simple to quite complex, but a simple voltmeter with an analogue gauge measuring voltage output is sufficient.
Check that the battery charger alligator clamps are free of corrosion or dirt that would inhibit an electrical charge from passing from the charger to the battery. Check the battery terminals for the same. If both are clean, the problem lies elsewhere.
Make sure the electrical socket into which the battery charger is plugged works properly. Check the lamp first by plugging it into an outlet that you know is operational to make certain the bulb isn't burnt out. If it doesn't work, change the bulb and try it again. Once you've verified the light is working, plug it into the outlet you will be using for the charger and turn it on. The light should work. If it doesn't, you've likely identified the source of the problem at the outlet itself, wiring to the outlet or at the fuse box.
If other outlets on the same electrical line aren't working, the circuit may have blown a fuse. If the rest of the outlets work, disable the fuse and make sure the wiring to the outlet you will be using for the charger is connected properly.
If the charger is battery-powered, as some are, test the charger battery by removing it and placing the positive end of the voltmeter probe or clamp on the positive end of the battery and the negative end of the voltmeter on the negative end of the battery. If the voltmeter shows the battery is fine, replace it in the charger. If not, you've likely identified the source of the problem and replace the old battery with a new one.
Plug the charger into the outlet that you have now verified as working. If the charger is battery-operated, replace the battery. Attach the positive end of the voltmeter to the positive lead from the charger and the negative lead from the voltmeter to the negative lead from the charger. If the voltmeter shows no reading, your battery charger is dead. If the voltmeter reads 12 or more volts, its working properly and the problem probably lies with the vehicle.
The problem with the charger, if it draws its power from an electrical socket, could be something as simple as a broken wire from the charger to the outlet. Check to make sure it isn't frayed or broken. If you're handy and so inclined, you may want to simply replace the wire and see if that fixes the charger. If the problem lies with the vehicle remember two things. Start with the simplest and least expensive possible problem and work your way to the more complicated and expensive possible problem, isolating each problem along the way. So start with the voltmeter and battery and work from there.
Always make certain you're connecting positive to positive and negative to negative leads (red to red, black to black) or you risk shock or explosion if working with the car battery.