Using PCM1807 with a microphone
It all depends on your application. You have two important matters to keep in mind: what you are going to do with that digital signal and how you are going to configure the ADC/retrieve data from ADC.

The DSP exists there to fulfill that first bit ("you have a digital signal - so what?"). You most likely want to process and interpret that signal in your application. The DSP is a typical application component.

The ADC you have presented (PCM1807) has a SPI interface and a MCU should be used for the necessary ADC configurations and commands - thorough datasheet reading is needed here. But now that you have a MCU there, the DSP existence depends on your desired application - you could be doing signal processing in that very same MCU, for example. This depends of course on your MCU capabilities.

The PLL is necessary as system clock for the proper signal sampling, as it needs to output a signal with frequency that is multiple (x256, x384 or x512) of sampling frequency.

Considering you are just starting out with ADCs (and 24 bits is just slightly an overkill), may I suggest a simpler chip? Something like MAX11647 has only 10 bits, 94.4 kHz sampling frequency, but has an overall simpler application circuit.