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