Its important to say you can farm the dungeon as you like, there is not only one correct way to do this. But i usually do this route. When you have completet the route run outside the dungeon and reset, then you can go in again. Max 10 times an hour. Remember to kill and loot everything!
A fast way to get out i using the druid Dreamwalk abillity, and then using it again will port you back outside the dungeon. Death Knight and Monks have i similar abillity called something else.
Blackfathom Deeps Drops
When you are farming this dungeon you are pretty much going for the transmog, which you can learn yourself ore try to sell on the auction house.
These mats have a change to drop at least if you run my route and kill/loot everything.
Slimy Murloc Scale
-Tangy Clam Meat
-Small Lustrous Pearl
If you have skinning, these mats have a change to drop.
There are not a lot of skinable mobs in this dungeon.