Here is the full list of all 27 Jack Axe achievements.
Defeat 6th Boss
Finish the Game
Shoot Tower 2 Cannon
Shoot Tower 4 Cannon
Shoot Tower 3 Cannon
Shoot Tower 1 Cannon
Hit a Jackpot on Lucky Wheel
Defeat 5th Boss
Repair the Platforms
Repair the Ship
Repair the Hanging Claws
Repair the Nut Screws
Defeat 4th Boss
Break the Underwater Glass
Help Mr. Barley Escape
Collect 50 Runes
Defeat 3rd Boss
Defeat 2nd Boss
Activate the Bridge
Help Cactwos Escape
Collect 20 Runes
Defeat 1st Boss
Last two minutes in Juggler
Last a minute in Juggler
Last 30 seconds in Juggler
Collect a Rune
Get the Axe