If this is a little bird house, probably the easiest way would be to make the roof as water tight as possible, then add the greenery. Hardware cloth can be manipulated into shape, lined with sphagnum moss and filled with dirt for planting. At the show, they might have been able to baby their plants a lot. So just remember that outdoors it will need some soil depth and lots of water in dry weather. Succulents probably would do well, thyme, epiphytes, or some grasses to feed the birds!
If you mean a larger aviary, think SUPPORT. A big box of dirt will make it very top heavy, so you need strong beams and reinforcement at the corners. I think I’d play it safe and just let some grapevines or hops (or roses!) grow up and over mine, like a pergola. Good luck!