Here is the full list of all 27 N++ trophies - there are 12 bronze, 7 silver, 7 gold and 1 platinum.
You are literally a ninja
Die 5000 times
Explore the Options menu
Beat the entire expert row of Solo Episodes
Beat the entire bottom row of Solo Episodes
Beat the entire middle row of Solo Episodes
Die in every possible way
Beat an entire column of Race Legacy episodes.
Kill another player using a rocket 5 times in a single MP Race episode.
Beat an entire column of Race episodes.
Beat an entire column of Co-op episodes.
Beat the entire top row of Solo Episodes
Beat an entire column of Solo Legacy episodes.
Beat an entire column of Co-op Legacy episodes.
Die 1000 times
Watch a highscore replay.
Triumph over adversity
Beat a Solo Episode from the bottom row without dying
Earn an All-Gold Badge in a Solo Episode from the bottom row
Beat a Solo Episode from the middle row without dying
Earn an All-Gold Badge in a Solo Episode from the middle row
Beat a Solo Episode from the top row without dying.
Earn an All-Gold Badge in a Solo Episode from the top row
Get all the gold in the Solo Introduction
Die by hitting the ceiling too hard
Die 100 times
Die during the solo introduction