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