If your ESC does not provide reverse functionality, it will never do.
The "reverse" mode can be achieved in general by swapping any two motor cables. I would do this with the arduino setup in combination with your R/C settings.
Arduino can easily get the RC information from a receiver.
Now it depends on your relays, you may achieve it with 2 relays, but if the relays are not the right ones you need 4 of them
To enable reverse, a popular design is to use an H-bridge (essentially a circuitry version of manually flipping the polarity of the wires) placed between your ESC and the motor. According to this answer, H-bridges are a normal part of an ESC output stage -- that your ESC appears to lack.
Although it could be built with relays, it may be better for your ESC and/or motors to select an appropriate semiconductor-based H-bridge. There are ICs available specifically for this purpose.