Here is the full list of all 47 The King of Fighters XIII trophies - there are 33 bronze, 11 silver, 2 gold and 1 platinum.
Unlock all trophies.
Clear 200 trials in Trial mode
Create 10 characters in Customize
Hit 100 times using EX Special Moves (Arcade, Versus)
Hit 100 times using EX Super Special Moves (Arcade, Versus)
Hit 100 times using NEO MAX Super Special Moves (Arcade, Versus)
Clear Time Attack mode with 30 characters
Play 300 [ranked matches / player matches]
Perform 300 Target Actions (Arcade)
Get 10 consecutive wins in a ranked match
Get 5 consecutive wins in a ranked match
Complete STORY.
Perform 50 Super Special Moves Finishes (Arcade, Versus)
Perform 50 Super Cancels (Arcade, Versus)
Perform 50 Drive Cancels (Arcade, Versus)
Perform 50 MAX Cancels (Arcade, Versus)
Hit 100 times using Super Special Moves (Arcade, Versus)
Get 50 NEO MAX Super Special Moves Finishes (Arcade, Versus)
Defeat 35 characters in Survival mode
Defeat 3 characters in Survival mode
Perform 10 Perfect Victories (Arcade, Versus)
Win 100 [ranked matches / player matches]
Get 2 consecutive wins in a ranked match
Play 100 [ranked matches / player matches]
Create 10 rooms
Win 50 [ranked matches / player matches]
Play 50 [ranked matches / player matches]
Win 25 [ranked matches / player matches]
Win 5 [ranked matches / player matches]
Complete the tutorial
Win your first [ranked match / player match]
Play your first [ranked match / player match]
Clear STORY.
Clear Time Attack mode for the first time
Get an SS at the victory screen
Perform 50 HD Cancels (Arcade, Versus)
Be challenged by "Saiki" and win in ARCADE
Get 10 straight wins (Arcade, Versus)
Register an icon, team and message in Customize
Complete all trials for 1 character in Trial mode
Clear ARCADE MODE on VERY HARD
Clear ARCADE MODE without continuing
Be challenged by "Ash" and win in ARCADE
Be challenged by "Billy" and win in ARCADE
Hit 100 times using Special Moves (Arcade, Versus)
Perform 10 Target Actions (Arcade)
Get an S at the victory screen