Here is the full list of all 21 Yu-Gi-Oh! Legacy of the Duelist achievements.
Perform a Fusion Summon
Perform a Tribute Summon
Defeat Classic Campaign
Defeat GX Campaign
Defeat 5Ds Campaign
Defeat ZEXAL Campaign
Perform a Xyz Summon
Perform a Synchro Summon
Perform a Pendulum Summon
Win an unranked multiplayer match (duel)
Win a ranked multiplayer match (duel)
Win a Battle Pack match (duel)
Unlock all avatars
Win all Duelist Challenges
Deal 200k in Direct Damage
Deal 100k in Effect Damage
Deal 50k in Reflective Damage
Chain 100 times
Create your first deck
Create 5 decks
Win a duel by having all Exodia cards in hand