To get this you must finish reading a chain of letters from all the main characters as well as the 4 filler characters that you get chain letters from. To start getting letters from other characters, you must first have a messenger bird unlocked. Next you have to fill at LEAST one heart on the character, you can do so by completing a request mission that involves that character and gives you a heart for them after completion or by giving that character an special item that fills up their heart gauge. For males you can give them a Ninjutsu Reference Book from the Item Shop and for females you can give them Flowers from Ino’s Flower shop (both cost 10,000 ryo a piece). Once you have at least one heart on that character, so long as that character isn’t in your support party then they will send you letters. After 3 letters you will have completed that character's correspondence event. You can do multiple characters at the same time.
For the 4 filler characters; after completing the main story a book merchant will appear right by the "Amagurina" at the front of the village and sell 4 different books at 500 ryo a piece. After buying one of the books you will receive letters from a random character relating to that book. The book merchant appears multiple times so you can buy each book, he also appears while you’re already doing a chain letter with the current filler character. He'll appear randomly, but don't stress over not getting him early, just keep working on your friendship with the main characters and viewing friendship events until he shows up.