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