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