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