Emblems are icons you unlock to place next to your gamertag or steam name. This is how to unlock each of them.

Emblem	How to unlock
BFF	Use 100 Co-op Moves
Boss Buster	Defeat 100 Bosses
Brainiac	Play a story mode Level (Single player) on Normal or harder difficulty and never switch out from Donatello or get knocked out
Bullet Rain	Defeat 30 enemies in a row using a turret
Can Crusher	Defeat Super Shredder without equipping any charms or using any Ninjutsu, items, or power ups.
Charmer	Collect all charm types
Chopper Chopper	Destroy 30 Helicopters
Clean Feet	Use a gimmick to travel 100 ft without touching the ground
Close Call	Defeat a boss while all players are in the pizza room
Defense Artist	Perform 100 Perfect Guards
Don't Need Help	Play a story mode Level (Single player) on Normal or harder difficulty and never switch out from Raphael or get knocked out
Economic and Efficient	Defeat 5 enemies at once with 1 Ninjutsu
Elite Challenger	Defeat 10 Elites
Elite Combatant	Defeat 50 Elites
Elite Conqueror	Defeat 100 Elites
Experienced blocker	Perform 50 Perfect Guards
Fearless	Complete all stages on Very Hard Difficulty
Field Doctor	Help Recover or Revive your buddies 50 times
Fledgling Ninja	Defeat 50 Enemies with Takedowns
Gotta go fast	Defeat 10 enemies with takedowns within 30 seconds
Hard Worker	Get an S rank 50 times
Hold Perfectly Still	Defeat 5 enemies during Turtle Time. (Enemies defeated by other Turtles also count)
Humiliator	Defeat General Krang with a Krangdroid
Kung Fu Master	Perform a parry (Perfect Guard) 5 times in a row on a boss
Maniac	Obtain all Collectibles
Mass Transit	Have 5 enemies get hit by a train at once
Master Warrior	Defeat 500 Enemies
Natural Leader	Play a story mode Level (Single player) on Normal or harder difficulty and never switch out from Leo or get knocked out
Ninjutsu Legend	Fully upgrade all ninjutsu
Ninjutsu Master	Obtain all Ninjutsu
Ninjutsu Novice	Obtain Half of all Ninjutsu
One Man Party	Play a story mode Level (Single player) on Normal or harder difficulty and never switch out from Mikey or get knocked out
Only the Best	Get an S Rank 100 Times
Perfect Synch	Perform a communication action during the snapshots when a boss is defeated
Professional Medic	Help Recover or Revive your buddies 100 Times
Running Circles	Perform a back attack (Perfect Dodge) 5 times in a row on a boss
Seen them all	Defeat all secret Bosses
Showoff	Have Mikey do his co-op move finishing pose during the snapshot when a boss is defeated
Skilled Warrior	Defeat 250 Enemies
Sky Cleaner	Destroy 15 helicopters
Slam Dunk	Use a heavy attack to knock an enemy against the basketball board in the Residential area, defeating him
Stealthy to the End	Defeat all bosses with a Takedown
Street Warrior	Defeat 50 Enemies
Turtle Power	Defeat Mega Krang without using a Krangdroid
Ultimate Ninja	Defeat 100 Enemies with Takedowns
Untouchable	Defeat Super Shredder on Hard mode without being shell-shocked once.