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