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