Here is the full list of all 76 Cave Story+ achievements.
No Life Capsules (Best Ending)
Completed Blood Stained Sanctuary in Less Than 3:00
Completed Blood Stained Sanctuary in Less Than 4:00
Completed Boss Attack in Less Than 8:00
Completed Boss Attack in Less Than 10:00
Completed Blood Stained Sanctuary
Completed Blood Stained Sanctuary in Less Than 5:00
Completed Blood Stained Sanctuary in Less Than 6:00
Completed Wind Fortress in Less Than 4:00
Completed Wind Fortress in Less Than 5:00
Completed Wind Fortress in Less Than 6:00
Completed Wind Fortress in Less Than 7:00
Defeated G-Clone!
Completed Boss Attack
Completed Boss Attack in Less Than 12:00
Completed Boss Attack in Less Than 15:00
No Life Capsules (Good Ending)
Completed Hard Mode
Defeated Ballos!
Defeated Heavy Press!
Saved Curly Brace
Obtained the Iron Bond!
Obtained Ma Pignon!
Obtained the 290 Counter!
Obtained the Alien Medal!
Obtained the Turbocharge!
For the rest of us!
Played Cave Story on April 29th!
Mask Mimiga Ending
Completed Easy Mode
Found and collected all Life Capsules
Obtained the Whimsical Star!
Obtained the Arms Barrier!
Normal Ending
Spared Eggfish
Defeated the Undead Core!
Defeated the Muscle Doctor!
Defeated the Doctor!
Defeated Misery!
Defeated the Red Ogre!
Found and collected all Missile upgrades
Obtained the Nemesis!
Returned Mister Little back home
Found Mister Little!
Bad Ending
Defeated the Twin Dragons!
Obtained the Spur!
Obtained the Bubbline!
Defeated Ironhead!
Defeated the Core!
Obtained the Tow Rope!
Obtained Curly's Panties!
Obtained the Machine Gun!
Obtained Chako's Lip Stick!
Beat by Toroko
Obtained the Super Missile Launcher!
Defeated Balrog, yet again...
Defeated Monster X!
Accumulated XX,XXX of damage
Defeated Puu Black!
Obtained the Snake!
Obtained the Blade!
Defeated Toroko...
Obtained the Life Pot!
Beat Omega with Jenka's Puppy
Defeated Omega!
Defeated Curly...
Defeated Balfrog!
Defeated Balrog, again...
Obtained the Fireball!
Defeated Igor!
Obtained the Missile Launcher!
Defeated Balrog!
Obtained the Map System!
Obtained the Polar Star!