Preheat the oven to 350 degrees. Grease a 13x9 inch pan.
In a large mixing bowl, beat the cream cheese until smooth. Beat in the sugar, lemon juice and lemon zest. Scrape mixture into a separate bowl and set aside.
In the same mixing bowl you used for the cream cheese mixture, combine the cake mix and butter. With a pastry cutter or two forks, mix until crumbly. Measure out 1 cup of this mixture and set side.
Add the milk and eggs to the mixing bowl mixture, and beat on low speed of mixer until mixture comes together. Increase speed and beat for 1 minute.
Spread mixture in the prepared baking dish.
Drop spoonfuls of the cream cheese mixture over the cake batter then use a knife to drag the puddles of cream cheese together to make one layer.
Sprinkle reserved crumb mixture over the cream cheese layer.
Bake until the edges of the cake are golden brown, and the top of the cake springs back when lightly pressed with your finger, 30 to 33 minutes.
Allow the cake to cool completely, then prepare the icing. Mix the confectioners' sugar and melted butter, then add lemon juice and just enough warm water so that the icing is about the consistency of glue. Drizzle it across top of the cake.