I would advise you to check the ignition system. The ignition system would include the battery, starter, carburetor, spark plugs, high-tension wires, distributor cap and those electrical components found in it.
Check the battery first. If it still good, check if the carburetor is putting gas in the engine. If it is, check if the high-tension wires are conducting electricity.
You will know if the alternator is not working if and when the battery won't charge.
