Can I build a camera that is sensitive at 2.4GHz?
Well, it has the potential to work.  You would have to line the inside with RF absorbing material, otherwise the incoming waves would just bounce all over the place.  

Using copper plates to detect the RF power probably isn't the best idea.  I would recommend using actual wifi antennas for that purpose, each one connected to an LNA and 2.4 GHz bandpass filter and crystal or diode detector.  

Another (probably better) option to consider would be a phased array setup. This is a bit more complicated, but you would not need the box or RF absorbing foam.  In this case, you would take an array of antennas (say, a 4x4, 8x8, or 16x16 grid) and connect them to a set of devices called Butler matrices.  A butler matrix is a type of passive beam forming network.  These devices consist of hybrid couplers and phase shifters arranged in such a way that they map distinct 'beams' out of the array to separate ports.  Basically, the idea is they act like a lens, except the focusing is done AFTER the signal gets captured by the antennas.  For a 4x4 grid of antennas, each butler matrix requires 4 hybrid couplers, and you would need 8 matrices - 4 for horizontal and 4 for vertical.  You're fortunate to be working at 2.4 GHz - it's possible to build reasonably-sized hybrid couplers at that frequency just in copper on a circuit board, making it possible to build a complete butler matrix on a single PC board, with no components aside from the connectors.  It would be possible to build 8 port or 16 port butler matricies (had to be a power of 2), though the larger the matrix, the more complicated it gets.  The outputs of these would then get passed through LNAs, 2.4 GHz bandpass filters, and crystal or diode detectors.  

Picture of butler array interconnection for an 8x8 antenna array: