Edit: With recent changes to quests in Nud-Melek and Zelem-Melek, I don’t think you even need to go here any more.
Balin’s Camp is located in Nud-Melek, midway along a corridor that runs between The Chamber of Mazarbul and The Deep Crossroad. Now, it looks accessible on the Durin’s Way map that comes with the game but it isn’t. The eastern end of the Chamber of Mazarbul is blocked off with rubble.
To get there you need to get to the eastern end of Nud-Melek, near where the passage from The Fanged Pit joins. Then you need to run a gauntlet through a variety of orcs and gredbyg to reach it. Assuming they’re of a level to attack you. Balin’s Camp is midway, and there are two things there to click for quest updates.