What the achievement asks
Reach Appreciated with Ve'nari, the broker who runs the only safe corner of The Maw. Ten points, account-wide, both factions.
It is the top rung of her ladder, which climbs Apprehensive, Tentative, Ambivalent, Cordial, Appreciative. There is no shortcut and no cap-raising item: this is a grind, and it is the one achievement that quietly gates most of the rest of the zone.
Why to do this one first
Ve'nari's standing is not a reward, it is the key. Her questline is what opens the closed parts of The Maw, and her stock is what makes moving around it bearable.
Half the zone is shut or hostile until you are further up this ladder. Explore The Maw, Better to Be Lucky Than Dead and Hunting Party all get easier the moment this one is done, so do it before them, not after.
Where the reputation comes from
You meet Ve'nari through the covenant campaign, and she then offers a chain of "Rule" quests that unlock the different areas of The Maw. Those come first, and they pay well.
After that the reliable sources are:
The daily quests, from Ve'nari herself and from around the zone.
The rares, all twenty-one of them, which are on your route anyway for Better to Be Lucky Than Dead.
The lore items, uncommon drops from Maw trash and rares. They are also the criteria for Deciphering Death's Intentions, so that achievement comes along free.
The weekly turn-in, The Cost of Death, which takes 800 Stygia off you and hands back a large block of reputation. It is the single biggest lump available, and it is the reason to bank Stygia rather than spend it the moment you have it.
The alt route
Ve'nari's reputation is warbound now, so gains on any character add to the same total, and Stygia moves between your characters through the warband bank.
That turns the weekly into the fastest path there is. A character taking every skip on offer can be through the campaign and standing in front of Ve'nari in well under an hour, and each one of them can hand in another 800 Stygia. The limit stops being time and becomes how much Stygia you can find.
The Stygia runs out faster than you expect. Take a different route through the zone on each character and kill the named rares and the trash on the way, rather than beelining for the turn-in.
What Appreciated gets you
The quest Rule 7: Betrayal is Inevitable, which closes out her story, and her last two items: the Possibility Matrix and the Spatial Realignment Apparatus.
It does not count as Exalted, so it will not move the counter on the 100 Exalted Reputations achievement or its siblings. That surprises people who grind it expecting both.