When the level loads, head to the right. You don't need to talk to Bofur, but you do need to get him some firewood... for some reason.
Once you grab the firewood, skip the following cutscene. Head across the ledges over the water and continue through the level. Once you've climbed the ropes, you can do a long-jump to the right of the save pedestal to hit a cutscene trigger. (Don't worry, you'll get warped back up once the cutscene starts).
Skip the cutscene and continue through the Stone-Giants. Once you reach the pillar, go around the main area to the right, along the ledges jutting out. This is to avoid a textbox, which stops rocks from falling while it's active.
Once the Stone-Giants knock down the pillar, continue across and climb the rope to the left. Jump to the ledge, then enter the next tunnel. Pass over the cave with the wolves, then drop down into the cave with the... frogs? Imps? Baby goblins? Whatever. Make sure to do a jump-attack before you land to reduce the amount of damage you take.
Head out of the cave using the exit to your right. Head along the wall to the left and enter the tunnel.
In the next cave, you'll find a machine that's missing several gears. You don't need to trigger the cutscene that tells you this; just enter the next room to collect all of the gears. Once you've gotten all of the gears, you'll be able to re-enter and put them in the machine, opening a door. Exit through the door.
Once you're past the first machine and back outside, you can do a cool-looking jump from the stairs:
Once you're past the stair jump, you need to do a semi-precise long-jump to skip the next 2 machines:
Once you've done the long-jump, pick up the key, open the door, and head through. Drop down into the small hole at the end of the cave, then continue down the tunnel. Follow the path outside until you reach the Dwarves.
Once you reach the Dwarves, you can do a slope-boost up the wall and interact with Thorin at the peak: