I've seen electrolysis using glass jars and baking soda to separate water to make hydrogen and oxygen using the alternator for additional fuel for gasoline engines. Is this likely to cause damage to the engine and if not, how do I make it work on a fuel injection engine? Is it dangerous in any other way?
Note that whatever hydrogen you produce will contain energy from the car battery. The car battery is charged by the alternator, which gives you energy obtained from fuel.
What you describe aims to be a perpetual motion machine but isn't. The inefficiency of the electrolysis (less than 100%) and the inefficiency of the alternator mean your overall scheme actually REDUCES the fuel efficiency of your car, not increasing it.
If you want to have a car run by electricity, consider buying an electric car. Way more energy-efficient than fiddling with hydrogen.