CPARKTX:
So what is the correct approach?
Learn enough electrics and electronics to safely hack into an electronic device and interface it to something it was not designed to interface with. There is no silver bullet here, and short of somebody basically doing the research for you and telling you step by step which wire to connect where I don't see any sensible way for you to do this safely.
Do you actually need to use the existing controller? In some ways it would be simpler if you could chuck that out and replace it with an Arduino and motor driver - but some of these really basic RC cars also have steering functions integrated into the controller circuit, and I can't see from here how the steering works.