"I don't have classes this afternoon." If you are referring to today OR "I don't have classes in the afternoon." If you generally don't have classes in the afternoon. There is no formal or informal way to say that. One phrase talks about today and the other talks about what is usually happening day to day.

Afternoon is the period between noon and early evening. He doesn't have to go to school during that time.

I don't have classes in the afternoon.