How can you make electricity without a battery?

You can make your own battery, or you can use a generator of some sort.

To make a battery, you need an electrolyte of some sort and electrodes from opposite ends of the period table. You would likely use zinc for one of the electrodes and copper or carbon for the other. You could use citrus fruit or potatoes for the electrolyte medium, and place the electrodes into them.

For a generator, you would need to create moving magnetic fields inside of coils of wire. You would have to create the movement at a steady rate, and use circuits to regulate the voltage and current produced.