There are 47 Trophies that can be collected.

Bronze

Let's see what you've got…
Clear STORY.

Show me what humans are made of…
Perform 10 Target Actions (Arcade)

Strength!
Hit 100 times using Special Moves (Arcade, Versus)

Spinning!
Hit 100 times using EX Special Moves (Arcade, Versus)

Yahoooo!
Hit 100 times using Super Special Moves (Arcade, Versus)

Yeeeeaaah!
Hit 100 times using EX Super Special Moves (Arcade, Versus)

Hey!
Hit 100 times using NEO MAX Super Special Moves (Arcade, Versus)

Yay! Perfect!
Perform 10 Perfect Victories (Arcade, Versus)

Hmph… this is only natural.
Get 10 straight wins (Arcade, Versus)

Play time is over!
Perform 50 Super Special Moves Finishes (Arcade, Versus)

And the final blow…!
Get 50 NEO MAX Super Special Moves Finishes (Arcade, Versus)

Come back later!
Perform 50 Drive Cancels (Arcade, Versus)

Mission Complete!
Perform 50 Super Cancels (Arcade, Versus)

Piece of cake!
Perform 50 HD Cancels (Arcade, Versus)

Heheh. Not bad.
Perform 50 MAX Cancels (Arcade, Versus)

Time will soon turn to ashes…
Get an S at the victory screen

After all, you're just trash.
Get an SS at the victory screen

You're not so bad!
Complete the tutorial

It's about time to start!
Register an icon, team and message in Customize

Here I come, buddy!
Create 10 characters in Customize

Hey, hey, hey!
Be challenged by "Billy" and win in ARCADE

Great
Be challenged by "Saiki" and win in ARCADE

You can't win against me!
Be challenged by "Ash" and win in ARCADE

Looks like I was just on time.
Clear Time Attack mode for the first time

I will execute my mission!
Defeat 3 characters in Survival mode

Let's play warrior!
Create 10 rooms

Go easy on me!
Play your first [ranked match / player match]

Good!
Play 50 [ranked matches / player matches]

Yeah!
Win your first [ranked match / player match]

This was just a greeting!
Win 5 [ranked matches / player matches]

Doesn't it feel good?
Win 25 [ranked matches / player matches]

Okay!
Get 2 consecutive wins in a ranked match

Number 1!
Get 5 consecutive wins in a ranked match

Silver

Do you understand now?
Complete STORY.

What's wrong?
Perform 300 Target Actions (Arcade)

How was it?
Clear ARCADE MODE without continuing

This is… my victory!
Clear ARCADE MODE on VERY HARD

You can't compare to me.
Clear Time Attack mode with 30 characters

I'm enough for this job!
Defeat 35 characters in Survival mode

That should do it!
Complete all trials for 1 character in Trial mode

Hehe… hot, wasn't it?
Play 100 [ranked matches / player matches]

I shall not waver!
Play 300 [ranked matches / player matches]

Isn't this fun, eh?
Win 50 [ranked matches / player matches]

Yeah! I did it!
Get 10 consecutive wins in a ranked match

Gold

Yes! I'm the best!
Clear 200 trials in Trial mode

Excellent!
Win 100 [ranked matches / player matches]

Platinum

THE KING OF FIGHTERS
Unlock all trophies.

http://www.ps3trophies.com/forums/ps3-trophy-news/6585-king-fighters-xiii-trophy-list.html