Cut the salmon filet into 4 pieces - leaving the skin on.
Salt and pepper the salmon to taste.
Sprinkle a little dill weed on each portion.
In a medium non-stick skillet, melt the butter over medium-high heat.
When the butter is melted and almost beginning to brown, add the fish, skin-side down.
Reduce the heat to medium.
Cook for 5-6 minutes, watching to be sure that the pan does not get to hot. You don't want the butter to burn.
While the fish is cooking, squeeze a generous amount of lemon juice over the fish.
After several minutes, you should see the fish beginning to change color as it cooks. When the fish is cooked about 1/2-3/4 of the way up, turn the fish over so that the skin is up - adding more butter as needed.
Cook for an additional 3 minutes. Turn the fish back over so that the skin is down. You are looking for the fish to turn opaque. Be careful not to over-cook! Fish is done when it flakes easily.
As soon as the fish is cooked, remove it from the pan to prevent over-cooking.
Serve with additional lemon juice and melted butter.