Is this the correct way to use this Audio AD Converter?
The outputs of the PCM1804 are a good first step. However, you should also look up the TOSLINK and (S/PDIF)/(AES53) interface specs, such as here and here and you'll see that the raw data is not enough. TOSLINK uses 32-bit packets, and only is guaranteed to support 20-bit data fields. Furthermore, audio fields are preceded by sync words, as specified in the AES link.

So, what you need to do is use the PCM1804 to digitize at a constant sample rate, and then use a microcontroller to buffer the data, assemble data packets, and then produce the bit stream which is sent to the transmitter.