Neon Abyss

When you earn enough gems (you get one for killing a boss at the end of each level) you can spend them at the bar on a skill tree. Once you've opened enough you'll get more trees, On the third tree, you can unlock a character called R-6, for 30 gems. This character only has shields, no hearts, so if you can complete a run with him then the achievement will pop. Doesn't matter it you try to pick up any hearts or get items that should give you hearts, you won't get any.

It's also easier to get the 10 shields achievement with this character too, just because you start with 3: -

In the fourth tree there is an item pack called Howard reactor , it will replace all hearts by shields , use it and complete a run , the achievement will pop .

I had a run that gave me a few items that gave me shields for doing certain actions, Like picking up coins or activating fast travel runes.
I put the game on easy, Then started the run avoiding hearts. After the first floor is when I started getting the shield items and by floor 3, I had 10 shields.
Every boss can be beat without taking damage, and if you watch a vid on youtube before fighting them you'll give yourself a better chance of making it to the end.
SinFluxx has added that picking up items that add heart containers doesn't void the achievement!