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