Snapchat only lists people as 'best friends' when you interact with them on a regular basis. The Snapchat best friends list is limited to three people, and is updated weekly.
Each week, Snapchat will look at who you send messages to the most, and will list the top 3 as your "best friends".
If you don't use Snapchat very often, or if you message a couple of people now and then, but no specific person very often, then Snapchat will display the message "No best friends" because it won't be able to work out how you message the most.
The best friends feature on Snapchat is really only a way to make it easier to find contacts that you communicate with on a regular basis. It doesn't mean that these people are actually your best friends in real life.
