Since you already have the screen you just need to add a texture to it.
Select the face(s) that make the computer screen and press ShiftD to make a duplicate. separate as a new object (Press P ->separate selection).
On the new object enter edit mode
UV unwrap it.
Create an emitter material that uses the video as texture an uses UV coordinates for the mapping.
Move the object to a new layer and create new render layer that contains only the layer with the screen object.
On the compositor import the image sequence you pre-rendered and use an alpha over node to superimpose the render layers that contain the screen and the .PNG sequence as background.
Other option (not necesarily easier or as precise) would be to track the 4 edges of the screen on the videoclip editor and create a plane track
see: https://blender.stackexchange.com/a/38332/1853 and Unable to "Create Plane Track"