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