Blackguard

It's important to know that you need to be EXALTED with the Thorium Brotherhood before you make the trip into BRD to buy the plans. :/

You can click the links under "Reagents" to get comments on individual mats, but here are the basics:

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• All of the items are tradeable, so you can farm them on an alt or buy them off the AH. You can also farm some of the base mats and have someone else craft the final mats.
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• Lava Cores and Fiery Cores are dropped by various mobs inside [zone=2717], and aren't very hard to get. You can get all you need in one run, but it might take two or three runs if you're unlucky. For both of these, as well as farming Dark Iron ore, you can repeatedly reset the instance as long as you don't kill any bosses, and get the mats in one sitting.
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• Guardian Stones are dropped by [npc=6560]s (easy trash mobs) in Un'Goro Crater. The only difficulty here is that you will probably have to wait for them to respawn 3 or 4 times, as there are just a couple mobs at a time. I farmed the mats for the Arcanite Bars while I waited for respawns.
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• Arcanite Bars are transmuted by alchemists. The Thorium Bars are smelted from Thorium Ore by miners, and are easy to farm in the Un'Goro Crater/Silithus area. The Arcane Crystals, however, take a bit of grinding. They are a rare drop from Thorium nodes, and cannot be found any other way. I had about 350 Thorium ore by the time I got 10 crystals, so it's not too bad of a grind, but does take a bit of time.
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• Dark Iron Bars are smelted from Dark Iron Ore at 8 ore to a bar. Since the cataclysm, all the world nodes have disappeared, so you have to farm the ore from Molten Core or Blackrock Depths (MC is the better choice here if you're like 80+ because there are far more nodes). If you're going to make the bars yourself, you have to get the Smelt Dark Iron ability, which requires 20 Gold Bars and 10 Truesilver Bars, both of which require rare ore nodes to farm, and 2 Star Ruby, which you will probably get by accident farming the other mats.
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The Black Forge, needed to smelt the Dark Iron Bars, is located near the very end of [zone=1584], just outside the entrance to [zone=2717]. The Black Anvil, needed to craft this sword, is located near the middle of the instance, on the platform [npc=9017] patrols. To get to either, go into BRD, turn right, click the mole machine, and hit "Just Past the Grim Guzzler". Now you're standing beside another mole machine in the middle of the instance.

If you look into the center of the room you're in, you'll see the Black Anvil, which you can jump down to from the mole machine, but if you need to smelt dark iron bars first, don't jump down yet. You'll save time if you clear to the Black Forge first.

From the mole machine, if you take the northwest hallway, then head past [npc=9156], a room filled with trash mobs, then through the room with the [object=169243] (you have to kill the guys to get through the next door), then take a right directly after the tomb with the Seven, you'll be at a bridge. To the left (north-east) of the bridge is the Black Forge.

You can also jump into the lava from the Black Anvil and just swim to the Black Forge, but then you can't get back to the Black Anvil without leaving the instance (if you do this, enter the Molten Core portal then run right back out, and you'll be back in the central area of Blackrock Mountain, where you can run back to the entrance of BRD). Since you probably need to hit the Forge before the Anvil, this is probably a bad idea.

You do indeed need to have your reputation with the Thorium Brotherhood at Exalted to buy ANYTHING from the vendor in BRD.