What the achievement asks
Complete the Blood Gate storyline in Zuldazar. Five points, Horde only.
This one is not tied to your account. It has to be completed by the character that runs it, so doing it on an alt does nothing for your main.
Where it sits in the war campaign
The Blood Gate is the northern pass out of Zuldazar, and this chain is part of the Horde war campaign rather than the Zuldazar zone story. It comes after the zone itself and follows the invasion outward.
It pairs with The Final Seal, which finishes the same thread with the Zandalar Forever scenario. Both are Horde only, and both are earned per character rather than per account.
Do them on the character you intend to keep
Because neither is account-wide, the sensible approach is to pick the character you actually play on the Horde side and run the whole war campaign on it, rather than splitting the chain across alts.
That character will also want Welcome to Zandalar behind it, since the war campaign does not start until the introduction is finished.