How can I use the TI ADS1243 8-channel ADC to convert a line-level audio signal (ideally balanced) to a digital 24-bit (as stated on data-sheet) stream.
You can't. The datasheet is surprisingly vague on this point, but this is a low-speed ADC, intended for instrumentation applications. The analog bandwidth is just a few tens of Hz at most.
That's well below what anyone would normally call audio.
The ADS1243 is intended for high resolution sampling of slow changing measurements - the datasheet mentions weight scales and blood analysis as intended uses.
Certainly nothing about using it in a stereo system.
There are multichannel 24 bit audio codecs out there. The ADS1243 isn't one of them, though.
The ADS1234 is a part designed for use in low bandwidth high resolution sensor sampling applications. In particular bridge style sensors are supported where gain is applied to the sensor reading and even 50/60 Hz filtering is performed. It indicates right in the "Features" section of the data sheet that signal sample rates of 10 or 80 samples per second are supported.
Audio sampling on the other hand is done at much higher rates. High performance audio sampling for CD is typically done at 44.1kHz and DVD at 48kHz. So as you can see, this device is not at all suitable for what you intend.
From the datasheet
When a 2.4576MHz crystal is used, the device can be programmed for an output data rate of 15Hz, 7.5Hz, or 3.75Hz. Under these conditions, the digital filter rejects both 50Hz and 60Hz interference.
With a maximum datarate of 15Hz, I think you can kiss audio goodbye