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