Here is the full list of all 33 N++ (NPLUSPLUS) achievements.
Beat 10 Hardcore Stories
Beat an entire Hardcore Story
Beat the entire expert row of Solo Episodes
Beat an entire column of Co-op episodes.
Kill another player using a rocket 5 times in a single MP Race episode.
Beat an entire column of Co-op Legacy episodes.
Beat the entire bottom row of Solo Episodes
Die 5000 times.
Beat the entire middle row of Solo Episodes
Beat an episode from the bottom row without dying.
Beat the entire top row of Solo Episodes
Earn an all-gold badge in an episode from the bottom row
Get all the gold in the Solo Introduction
Explore the options menu.
Beat an entire column of Solo Legacy episodes.
Beat an entire column of Race Legacy episodes.
Beat an episode from the middle row without dying.
Beat an entire column of Race episodes.
Die in every possible way.
Die by hitting the ceiling too hard.
Watch a highscore replay.
Earn an all-gold badge in an episode from the middle row
Beat an episode from the top row without dying.
Triumph over adversity.
Earn an all-gold badge in an episode from the top row
Die 1000 times.
Die 100 times.
Die during the solo introduction
Complete All Hardcore Stories
Get all of the All-Gold badges in every page of Solo
Get all of the All-gold badges in every page of secret levels
Get everything in Solo mode
Get everything in the Ultimate Edition of the game