Here is the full list of all 54 Sword Art Online Re: Hollow Fragment achievements.
Game Cleared with play time exceeding 100 Hours
Beat Yuuki
Defeat the final boss in floor 100
Defeat the boss in floor 99
Defeat the boss in floor 98
Defeat the boss in floor 97
Defeat the boss in floor 96
Defeat the boss in floor 95
Defeat the boss in floor 94
Defeat the boss in floor 93
Save 100 million col
Defeat the boss in floor 92
Defeat the boss in floor 91
Defeat the boss in floor 90
Defeat the boss in floor 89
Defeat the boss in floor 88
Raise your bond with Silica to the max level
Raise your bond with all heroines to the max level
Defeat the boss in floor 87
Defeat the boss in floor 86
Raise your bond with Philia to the max level
Defeat the boss in floor 85
Defeat the boss in floor 84
Raise your bond with Sinon to the max level
Defeat the boss in floor 83
Defeat the boss in floor 82
Raise your bond with Leafa to the max level
Raise your bond with Lisbeth to the max level
Defeat the boss in floor 81
Beat Hollow Kirito
Defeat the boss of the Control Area
Defeat the boss of the Otherworld Area
Defeat the boss of the Great Void Area
Beat PoH
Raise your bond with Asuna to the max level
Defeat the boss of the Cove Area
Defeat the boss of the Floating Rock Area
Defeat the boss in floor 80
Defeat the boss in floor 79
Defeat the boss in floor 78
Defeat the boss of the Sea of Trees Area
Defeat the boss in floor 77
Defeat the boss in floor 76
Complete tutorial
Unlock all achievements
Get every Last Attacking Bonus
Complete the game without any one of the lead group dying
Complete Hollow Missions
Complete 100 Implements
Upgrade weapons 200 times
Complete Gallery Lists
Kirito reaches at least level 150
Multiplayer Mode playtime exceeds 20 Hours
In a multiplayer game, defeat at least 100 HNM creatures.