To unlock this achievement you need one piece of mail from every mail sender.
Out of the 102 mails you only need 19 and the list is as follows:
SOLDIER (2) (All Story)
Kunsel: Congrats on your first mail (Story)
Luxiere: From Luxiere (Story)
Shinra News (1) (Story)
Shinra News: Midgar Construction Project
Lazard (1) (Story)
Lazard: SOLDIER Responsibilities
Friends (4) (Three Story, One missable)
Cissnei: You're not busy, are you?
Cloud: I wanted to tell you
Tifa: Is he alright?
Wonder Hunter: The Last Wonder (MISSABLE, refer to Seven Wonders Expert achievement guide:)
Shinra Truths (1) (MISSABLE)
In Chapter 5 speak to the reporter in Sector 8-Fountain and do not report him; let him go.
Shinra Truths: The Truth about SOLDIER!
Keepers (1) (MISSABLE)
Keepers of Honor: Angeal Fan Club Newsletter 311
Genesis Girls (2) (MISSABLE)
Red Leather: Genesis Fan Club Newsletter 666
Study Group: An Invitation to LOVELESS
Hero News (2) (MISSABLE)
Silver Elite: The scent of silver winds
Zack Fan Club: Now accepting applications
For fan club enrollment refer to the Fan Club Afficionado achievement guide:
Spam (3) (Unlockable, not missable)
These are unlocked by progressing Yuffie's mission line. Beating Bahamut Fury gives you the final mail necessary.
Yuffie's missions are the 8-4, 8-5 and 8-6 lines but you only need to complete through 8-5-6 for the achievement.
Treasure Princess: Recruitment
Anonymous: Just between you and me
Yuffie: Notice
Shop News (1) (MISSABLE)
In chapter 5, speak to the man who is talking to a woman in the Shinra Building entrance.
This unlocks 6-2-1: Slum Development Plan 1. Complete the mission to unlock a store and this mail.
Shop News: The Best shop in the world
Tutorial (1)
There are 5 Tutorial categories but the sender is always Tutorial so technically, you should only need
Tutorial: Controls.
These are all unlocked during the story so you can't miss them.
The earliest you can unlock this achievement is right after completing the Nibelheim seven wonders.