What the achievement asks
Complete the Summons from the Depths scenario ten times. Ten points, account-wide, both factions.
It runs on a clock, not on a queue
This is not something you can start. The event fires every three hours, always in the same place, and if you are not there you wait for the next one.
Ten completions therefore spread over at least thirty hours of real time, and in practice several days. It is the most calendar-bound achievement in Nazjatar.
Every completion counts. There is no daily or weekly cap, so if you can be present for two or three in a day, all of them tick. That is what shortens this from ten days to three or four.
What happens
An Azsh'ari Leviathan is summoned at the coordinate above and the players present bring it down. At current levels it dies quickly and there is nothing to learn.
The only thing that matters is being on the spot when it starts.
Fitting it around the rest
Set the times you can make and treat everything else in Nazjatar as what you do between them. The zone has plenty that fills three hours: the daily quests for Subaquatic Support, the world quest champions for Nazjatarget Eliminated, or a scrying run for Nothing To Scry About.
The one thing not to do is log in specifically for the event and log out after. Ten separate trips for a five-minute fight is how this becomes the achievement people abandon.