This level has 2 loading screens, with a cutscene in between. This is one of those rare cases where it's okay to spam START to skip, since there's a loading screen directly after.
Once you're past the 2nd loading screen, head to the left. You can do a long-jump to get across to the gate, which you can hop over with a simple jump. Once you're past the gate, head forward and climb the ledge, then head to the left to grab a Deep Cellar Key. Now head back and toward the main door.
Once you've gotten inside, you can skip getting one of the crystals for the door:
After that, head down the tunnel into the dungeon. You can climb up some barrels on the left, then head to the left to get to the Dwarves. Talk to them, then head back toward the barrel room. Enter the barrel room, trigger the cutscene, then immediately go back to the Dwarves.
Talking to the Dwarves will trigger a cutscene, which will warp you in front of a tunnel. Head up the tunnel, which leads outside. Grab the rope and hold on to it for an extra second to make sure it lowers, then jump off to the right. Climb up the brown rope, then jump to the small platform sticking up. Jump to the yellow rope from there, then to the door to the right. Grab the Deep Cellar key outside before entering.
Once you enter, put on your Ring and head forward. There's a web in front of you. You can't cut it, but you can burn it with your web potion. Burn the web, enter the room to grab the Deep Cellar Key, then turn around and head back outside.
Once you've gotten the Deep Cellar Key from the spider-cave, you should have 3 of the 4 Deep Cellar Keys. Once you do, head back down the tunnel toward the barrel room. You can do a cool clipwarp from here to skip the basement section of the level: