What the achievement asks
Obtain the Heart of Azeroth. Ten points.
This one is not tied to your account. The neck belongs to the character that receives it, and so does the achievement.
It is the first thing Battle for Azeroth gives you
Magni Bronzebeard hands the Heart of Azeroth over during the expansion's opening quests, before you have chosen a continent or a zone. Nothing has to be farmed and nothing can be missed.
The only characters that do not have it are ones that were levelled through a different expansion and brought to Kul Tiras or Zandalar without ever starting the Battle for Azeroth introduction.
If you are one of those, you do not need to replay the expansion. Going to Magni and accepting the neck is enough on its own, and it is worth doing even outside this achievement.
Why it matters elsewhere
The Heart is a prerequisite hiding behind several unrelated achievements, and its absence is the cause of some genuinely baffling problems.
The clearest example is Adventurer of Drustvar, where three rares refuse to give kill credit to a character that is not wearing the Heart. People kill them repeatedly, conclude the achievement is bugged, and the answer is a neck sitting in Magni's hand.
Everything else in the Heart category follows from here as well: The Heart Forge, Power Is Beautiful and My Heart Container is Full all need the neck first.