Paul is Alexander's buddy. He doesn't get too much exposition as a character, but he does serve the important purpose of highlighting the importance of social life for children.
Paul gets the praise for his sailboat drawing; he gets the best dessert at lunchtime; he gets to dictate who's friends and who's not. It's all pretty typical little kid schoolyard drama, but that doesn't make it any easier for Alexander to deal him him.
In other words, Paul's a stupid face.