the best idea would be to dismember the PC by components and wrapping them in bubblewrap and something that doesn't scratch the surface of the components.

Carrying the PC on the plane is not that easy.

the monitor and peripherals are not that much of a problem, but the Case is a big issue...

the components have to be placed somewhere in your luggage where they wont move around at the first turn or twist (you know how the luggage is handled).

one option would be to open the case, remove the HDD, optic drive, graphic card, memory, fans & coolers and PSU. Wrap them individually and put them back in the PC case without damaging the motherboard and make sure that they are in a tight fit so that they don't move.

The motherboard can be left mounted to the case with the CPU however it would be a good idea to remove the cooler and wrap it.