What's the best way to make a bully understand how horrible his behaviour is?

There is not a surefire way, unfortunately. You need to talk to a trusted adult or someone close to the bully and let them know of the effects.

Try to explain how hurt it is when you get made fun of, or try to show him and sooner or later he will understand that his horrible behaviour is hurting everyone

I don't personally believe bullies can be 'made' to understand how horrible their behavior is, until they are placed on the receiving end. The problem with that is, most bullies are bullies because they have already been in the same situation, and this is their way of avoiding ever being a victim again. I'm not saying it's right. But 9 times out of 10 there is a far deeper reason for someone becoming a bully. And until the underlying issue is unpacked, and people get to heal, a bully will remain a bully. So, instead of the usual fighting back or doing to them as they do onto you (which doesnt solve anything), try understanding what happened in their lives to make them this way. Its the hardest thing in the world to do, but the only way to get through. A bully may need counseling - there's a thought.