DC Motor Reverse and Overall Advice

I assume that your question "how can I build a circuit such as the one here onto that chasis?" Centers around how to add such a circuit without using the prototype board. The best choice would be stripboard.

Something like: 8019 Vector Electronics | Prototyping, Fabrication Products | DigiKey

Or: http://www.digikey.com/product-detail/en/adafruit-industries-llc/1608/1528-1101-ND/5154676