As long as your original Level 1 Character entered the Tristram Portal with 0 XP, you should be able to use that same Character.
Just make sure that once you start Labyrinth Level 1 you do not leave the game until you’ve completed all 16 Levels and killed The Dark Lord.
You did not specify which Achievement would not unlock. If you are trying to get all of the Achievements and their rewards, you might want to take a look at [SPOILER] The Darkening of Tristram Guide.
Good luck and good hunting! 