SQL tables for a chat app

What you are trying to do is quite simple for people familiar with the basics of databases but it is not so easy for people that are familiar with databases to explain to someone else that is not familiar with the basics of databases. You can really benefit from reading a book about databases.
 
What you can do is to create a table of users and a separate table of friends and you need to "relate" the two. Instead of me writing an explanation of how to relate the tables, it would be better for you to read a book or article about that.