The Best Friends list on Snapchat is the contacts you interact with the most. The top contacts on the Chats or Snap send screen automatically appear as Best Friends. You can't change this manually, as it updates based on activity.
To see your Snapchat Best Friends list, open Snapchat and swipe right to access the chat screen. Your top friends appear based on interaction frequency. Note that Snapchat no longer shows the full Best Friends list publicly for privacy reasons.