Here is the full list of all 76 Cave Story+ achievements.
Obtained the Fireball!
Obtained the Bubbline!
Obtained the Missile Launcher!
Obtained the Super Missile Launcher!
Obtained the Machine Gun!
Obtained the Snake!
Obtained the Spur!
Obtained the Blade!
Obtained the Map System!
Obtained the Life Pot!
Obtained the Tow Rope!
Obtained the Whimsical Star!
Obtained the Arms Barrier!
Obtained the 290 Counter!
Obtained the Turbocharge!
Obtained Ma Pignon!
Obtained the Iron Bond!
Defeated Balrog!
Defeated Igor!
Defeated Balrog, again...
Defeated Balfrog!
Defeated Curly...
Defeated Omega!
Defeated Toroko...
Defeated Puu Black!
Defeated Monster X!
Defeated Balrog, yet again...
Defeated the Core!
Defeated Ironhead!
Defeated the Twin Dragons!
Defeated the Red Ogre!
Defeated Misery!
Defeated the Doctor!
Defeated the Muscle Doctor!
Defeated the Undead Core!
Defeated G-Clone!
Defeated Heavy Press!
Defeated Ballos!
Normal Ending
Completed Boss Attack
Beat Omega with Jenka's Puppy
Saved Curly Brace
Found and collected all Missile upgrades
Found and collected all Life Capsules
Accumulated XX,XXX of damage
Obtained the Polar Star!
Obtained the Nemesis!
Obtained Chako's Lip Stick!
Obtained Curly's Panties!
Obtained the Alien Medal!
Found Mister Little!
Bad Ending
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 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!