Here is the full list of all 40 Accel World VS Sword Art Online Deluxe Edition achievements.
Cleared "The Impatient Skies."
Cleared a Sudden Quest.
Cleared an Extra Quest.
Defeated the aura-clad monster general.
Cleared a Challenge Quest.
Cleared a Friend Quest.
Cleared "The Invader."
Performed a Switch.
Obtained all achievements.
Cleared "Cold Trap."
Cleared "Disastrous Moonlight."
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.
Cleared an Extra Quest with only Kirito and Silver Crow.
Cleared an Extra Quest with only Asuna and Black Lotus.
Cleared an Extra Quest with only Sky Raker, Ardor Maiden, and Aqua Current.
Completed all of Hexe's requests.
Employed a blacksmith's services.
Destroyed a solid Otherworld object.
Used BP to buy something.
Raised one character's level to 500.
Raised one character's level to 1000.
Raised one SAO character's skill level to 500.
Raised one AW character's skill level to 500.
Enhanced one weapon to +10.
Raised a weapon from Rank 1 to Rank 10.
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.