Here is the full list of all 76 Cave Story+ achievements.
Found and collected all Life Capsules
Obtained the Snake!
Obtained the Bubbline!
Defeated the Core!
Defeated Ironhead!
Defeated the Twin Dragons!
Defeated Misery!
Defeated the Doctor!
Defeated the Muscle Doctor!
Defeated the Undead Core!
Bad Ending
Normal Ending
Completed Easy Mode
Found and collected all Missile upgrades
Obtained the Turbocharge!
Obtained the Super Missile Launcher!
Defeated Balrog, yet again...
Defeated Monster X!
Defeated Puu Black!
Obtained the Arms Barrier!
Obtained the Blade!
Defeated Toroko...
Accumulated XX,XXX of damage
Obtained the Life Pot!
Obtained the Machine Gun!
Defeated Balfrog!
Defeated Curly...
Defeated Omega!
Beat Omega with Jenka's Puppy
Obtained the Fireball!
Defeated Igor!
Defeated Balrog, again...
Obtained the Missile Launcher!
Defeated Balrog!
Obtained the Map System!
Obtained the Polar Star!
For the rest of us!
Obtained the Spur!
Obtained the Nemesis!
Obtained Chako's Lip Stick!
Obtained the Tow Rope!
Obtained Curly's Panties!
Obtained the Whimsical Star!
Obtained the Alien Medal!
Obtained the 290 Counter!
Found Mister Little!
Obtained Ma Pignon!
Obtained the Iron Bond!
Defeated the Red Ogre!
Defeated G-Clone!
Defeated Heavy Press!
Defeated Ballos!
Mask Mimiga Ending
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
Played Cave Story on April 29th!
Spared Eggfish
Saved Curly Brace
Returned Mister Little back home
Beat by Toroko