Start out by juicing your lemons until you reach about ½ cup of juice. Mix with water in a large pitcher and set aside. Place cubed watermelon and sugar in blender and pulse until it is completely liquified and sugar is dissolved. Strain watermelon juice through a coffee filter on top of a fine mesh sieve. This will remove all pulp from the juice. If you like pulp, feel free to leave out the coffee filter and just strain through the mesh sieve to only remove some. Once watermelon juice is strained, pour it into the lemonade mixture. Serve immediately.