So far I have only got as far as removing the concrete slab and what seemed like endless amounts of soil / infill within the dwarf walls beneath it, down to ground level (approx. 1m below?). Directly beneath the floorboards that were getting wet there was a joist or two supported by pieces of wood, which in turn sat on top of the dwarf wall. The theory is that moisture was coming up from the ground, travelling up through the soil and dwarf wall, and ultimately making the wooden supports and joists on top of it wet. On top of the dwarf i.e. separating the wood from the wall, I could see what i think was the old damp proof slate / felt that was designed to stop the water passing through but that had clearly long since failed.
I haven't yet put the damp proof membrane in but have used some plastic sheeting (....empty cereal packets) between the wall and the wooden supports to test if this would stop the water movement. It would appear to as the joists no longer look or feel wet and the floorboards remain dry. Previously the joists were so soft I could just push a screwdriver into them.
Our situation might be slightly different to yours in that the concrete hearth was the leftovers of a previously removed chimney breast. For us it was fine to break it up and remove it as the plan was always to extend the floorboards up to the wall. It would be slightly different if you wanted to keep the concrete hearth in place and so couldn't remove the soil beneath it. I will probably need to do that on our other fireplace but one job at a time....