First of all, this is a math game called picross or Nanogram (https://en.wikipedia.org/wiki/Nonogram). You may want to check some basic introduction and skills on Wikipedia or any content on the web.

Some general hints which can help you solve faster are

• try to find big numbers first (or a big sum of a column or a row)
  
• try to fill boxes on border rows or columns as soon as possible
  
• check if the picture is left-right symmetric (or almost symmetric) by observing the numbers, if so, you can solve from the center column (or the 2 center columns)

For achievements, you can get each achievement upon finishing each chapter. Depending on your skills, it can take around 30-60 hours to finish the game. If you wonder why some players have all achievements unlocked just within minutes, that was probably because they played a version where the dev had mistakenly unlocked all achievements. Days after, the dev fixed this bug.