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