What are the requirements to have an acheivement pop?
As stated in the description of almost any achievement:
"[insert task here] in quick or competitive play"
So no, achievements can only be earned if you are playing in quick or competitive. Since you don't get XP in Private games, you can't get the player level achievements that way but you can get them if you play vs.AI, which would take a lot more time than playing in Quick or Competitive.