There is some confusion around this achievement. Given the potential time investment, many people are unwilling to experiment too much. There were also some bugs with it on the PTR, so some of the existing information is slightly out of date.
Can I use an existing level 1 character? Does it have to be a new one?
It appears that the character must have been newly created after the event starts. Some players report that if they use preexisting level 1 characters, they do not receive credit.
What if I hit level 2 before entering the Labyrinth?
This is not a problem - the character must be level 1 when entering the game. I had some overpowered cube effects on my character and AOE-killed some mobs in Old Tristram on the way to the portal and hit level 2. I still received credit.
Must I complete the achievement in one sitting? What if I get disconnected?
Yes. If you are disconnected, either offline from Battle.net or completely kicked out of the game, you will not receive credit unless this occurs before you hit level 2. When your character reenters the game it will not be level 1, and will not meet that requirement.
Can I give my new character any gear? Paragon points?
Absolutely. Many players use the Gem of Ease and Kanai's Cube to give level 1 characters level 70 gear, even full sets. Hellfire jewelry is relatively easy to craft and gives a huge boost since it has level 70 stats but no level requirement. Paragon points are fine, too: it is an easy way to give a good boost in a primary stat.
Note: if you craft any gear (e.g. Hellfire jewelry), be sure to craft it on the level 1 character or another character of the same class. Otherwise, you run the risk of junk stats. For example, crafting the Hellfire Amulet of Strength may give wizard ability bonuses if crafted on a wizard, despite being targeted at a barbarian or crusader.
Can I port out of the dungeon?
Yes, feel free to portal back to town to empty your inventory.