It's really "janky" to use an alternative or external GPU without Thunderbolt or an Expresscard slot as you'll have to wire a cord underneath your laptop with the plastic cover off like what "hotrob" said. Also, in most cases you MUST use an external monitor because its too difficult to have signals going out to the GPU, than back in to the internal display. You'll lose some performance, normally due to a data bottleneck.
The G53SX should have a replaceable GPU in MXM (its what the GPU and the board are called) format, however, ASUS is weird and doesn't use standard MXM so you'll have to do some research to see if one would fit. You would also probably need a bios update to get it to work and even then there is no guarantee.
Honestly, you're best bet may just be to sell it for some compensation and buy a newer model, that can be fully 100% upgraded so you're never really "stuck".
TL

R its doable, but it wont look clean and it will be difficult. It's also expensive and could mitigate the performance gains by how much you could have put into a new system.
P.S. Interesting note, it's amazing how far "integrated" graphics have come. Intel's HD 530 (desktop version) performs similarly to your GTX 560M today, crazy huh!?