Here is the full list of all 40 Accel World VS Sword Art Online Deluxe Edition achievements.
Raised a weapon from Rank 1 to Rank 10.
Enhanced one weapon to +10.
Employed a blacksmith's services.
Cleared an Extra Quest with only Sky Raker, Ardor Maiden, and Aqua Current.
Cleared an Extra Quest with only Asuna and Black Lotus.
Cleared an Extra Quest.
Cleared an Extra Quest with only Kirito and Silver Crow.
Cleared a Challenge Quest.
Cleared a Sudden Quest.
Used BP to buy something.
Cleared "Cold Trap."
Cleared "Disastrous Moonlight."
Cleared "The Impatient Skies."
Cleared a Friend Quest.
Raised one character's level to 500.
Raised one character's level to 1000.
Raised one AW character's skill level to 500.
Raised one SAO character's skill level to 500.
Cleared "The Invader."
Defeated the aura-clad monster general.
Performed a Switch.
Obtained all achievements.
Cleared "Showdown."
Cleared "Lost Wish, Tumultuous Heart."
Cleared "A Gleam of Hope."
Cleared "Millennium Twilight."
Cleared "Timeless Bonds."
Cleared every "Navigation" episode.
Cleared every "Mystification" episode.
Cleared 50 quests.
Completed all of Hexe's requests.
Destroyed a solid Otherworld object.
Obtained 100 different types of equipment.
Received 50 titles.
Acquired 100 total titles.
Chained together 8 skills with Skill Connect.
Started episode 5 of "Say It, Miss Strea!"
Unlocked Seven.
Unlocked Nitride Unica.
Unlocked all seven Kings.