Here is the full list of all 76 Cave Story+ achievements.
Completed Boss Attack in Less Than 8:00
Returned Mister Little back home
Bad Ending
No Life Capsules (Good Ending)
No Life Capsules (Best Ending)
Completed Hard Mode
Completed Blood Stained Sanctuary in Less Than 3:00
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 Boss Attack
Completed Boss Attack in Less Than 15:00
Completed Wind Fortress in Less Than 5:00
Completed Wind Fortress in Less Than 6:00
Completed Wind Fortress in Less Than 7:00
Defeated G-Clone!
Completed Blood Stained Sanctuary
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
Played Cave Story on April 29th!
For the rest of us!
Defeated Ballos!
Defeated Heavy Press!
Mask Mimiga Ending
Completed Easy Mode
Spared Eggfish
Obtained the Turbocharge!
Beat Omega with Jenka's Puppy
Obtained the Snake!
Obtained the Machine Gun!
Defeated Balrog!
Beat by Toroko
Normal Ending
Found and collected all Missile upgrades
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 290 Counter!
Obtained the Nemesis!
Found Mister Little!
Defeated the Twin Dragons!
Obtained the Arms Barrier!
Obtained the Whimsical Star!
Obtained the Spur!
Obtained the Alien Medal!
Defeated Ironhead!
Defeated the Core!
Accumulated XX,XXX of damage
Obtained the Tow Rope!
Obtained the Super Missile Launcher!
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 Missile Launcher!
Obtained the Map System!
Obtained the Polar Star!