Spelunking the Den

Grab the climbing backpack and look straight up. You'll see a shiny grappling hook. Click it.

Face north. Now head WEST towards the first stalactite you can grab. Keep clicking stalactites until you've gone around to the other side and started grabbing onto wall rocks. Make your way around, and start looking for the bag with the key. Get close enough and click it.

Now drop down to floor, use the same grappling hook again, but this time, head EAST, clicking stalactites until you get to the climbing rocks. Click the climbing rock on the opposite side of the wall you're on for the last step, the look back towards the other side where you'll see a chest and a new grappling hook. Click that new hook, but DO NOT drop off. Click the chest while you're on the grappling hook to complete the quest.

HINT 1: Do NOT use 1st person camera, for some reason some of the stalactites clicks do not register in 1st person camera mode.
HINT 2: Watch out for the bat patrolling around as you click around.

SPELUNKING IN TWO CHUNKS
Part Zero - Where Is The Climbing Gear? And The Cave In General
/way 43.8 60.8 Spelunking Den with rare and climb quest

The cave entrance is decently huge, and is not on the very top of the mountain, basically about a quarter of the way down from that.

As you come in the giant cave entry, there's a little backpack behind a wide pillar at floor level.

The slyverns are sparse and relatively OK if you only have to fight one at a time.
Bad news, THIS AREA CAN PROC AN ELEMENTAL RARE ([npc=200681]) and they are a mean one with floor damage and so on.
Good news, if it pops a bunch of people will show up to help.
Great news(?), it drops loot worthy of a heroic grade fight, including BoA gear tokens for catch-up iLvl 385.

Part One - Getting the Key To Use Here
Not to be confused with the epic'ly useful Zskera Vault Keys, this one is just a regular quest-item key in a tiny pouch hanging off a rock. In the Style of extra loot in climb quests it hangs just below a reachable rock.

Ugh, you are about to ask, what rock, I can't tell where to start climbing. LOOK UP. Til your neck hurts virtually. There's a grapple hook there (somewhat square, hanging off the ceiling).
(credit: ph0tek, bigtimeweenfan.)

Position yourself with your back to the climbing gear and your face to it; you need line of sight to grapple. That fact becomes important in the end of Part Two.

Be ready to use your "prevent hazard" button, (button 1 in your climbing gear, thanks Vaelor for asking) the slyvern breath weapon is a fluffy white swirl in midair and not very big, but it will detach you from the stalactites or rocks. You won't need Rupert's hand.

Your vehicle-bar for this includes various Dragonscale Expedition features. when you mouse over them they have a tooltip for their features, but people with alternate-mice methods might not easily get that info.

Button 1 seems assigned to some sort of "hang on real tight" feature which in a variety of climbing quests is used to give you one evasion of a hazard.

So you can press 1 if you see the swirl of white, or maybe try hitting it when you see a baby slyvern flying around, as the effect is a teeny-tiny breath weapon from them.

Grab that hook. A long rope will carry you up there. grab-hand stalactites - and eventually rocks - to the clockwise direction (your right initially, if you faced as I said, but you might have turned your camera around to feel like you can see what you grab). There's rock, stalactite, rock again. GRAB THE LITTLE SACK, it contains the key. Don't hearth, don't quit, don't leave.

Fall off. Re-equip the Climbing Gear.

Ignore the spiral stair-ramp, it has no value to this quest.
The chest that is the prize is in an extra nook up there.

Part Two - Claim Your Prize
Look up again. Grab that hook again. Go the other direction.

You are gonna be somewhat "spider man" using your climbing-gear hand between the full size chunks of ceiling all these stalactites or rocks are attached to. WHEN YOU CROSS THIN AIR THIS WAY THE SECOND TIME there are two rocks and it looks like you're soooooo close. You can see that second hook off the top of the cavern, taunting you.

Click the rock that looks nearest it. Grab the rock to its right that seems farther from it, or slightly higher and therefore maybe the same distance? Its line of sight is broken and you cannot grab the hook from there. Turn your camera; there is another rock just behind your shoulder, on the rock across air. Grab that. Turn your camera the other way.
(credit: Talionus "on the opposite wall", avm1997 "it is not intuitive", Shanaraen "last one look back")
NOW YOUR LINE OF SIGHT TO THE SECOND HOOK WORKS.

Once you've grappled to that, while hanging from here you can cursor the chest that sits in that nook.

Edit to add: Unless you're in a Raid Group. Then you can't, until you get out of the raid. Thanks Raima

Edit to add: (thanks ArashiFaustes) Oh my, now I'm stuck in climbing gear, how do I get out? It counts as a Buff. You can remove it the same way you remove a Buff which is normally to right-click the buff icon in your list of buffs.

TL;DR get used to changing your camera angle. :(

Blizzard finally came up with a climbing quest that made me give up on it.