Line, Neutral and Ground and another Ground
Also I think (correct me if I am wrong) one should not connect the
  Neutral from the line to the ground from a PC.


Right. Never connect neither the phase nor the neutral of the mains to any node of a PC. If you did that, touching any node (including the chassis) of the PC could kill you.


  Fig 21 on page 54 of the datasheet is then plain wrong?


It is not wrong. It just assumes that all nodes around the CS5490 and the "Application Processor" (including the "triangular" ground) are electrically isolated from any end user. This means that, if you want to connect a PC to the CS5490, you MUST insert isolation between them. Easiest: insert isolation at digital links like RX and TX (isolating analog links is much more complex), using either optocouplers or magnetic isolators like these ones (I've used many of them, and they work very well. They allow you to work at higher frequencies than with usual optocouplers. Some of them even allow you to transfer power to the other side (the isolated one). Look for "isoPower" in that same page).


  Can I do it without the need of adding another power supply?


You need a power supply for the CS5490, which cannot be the same as your PC's power supply.


  And what does a pointing down triangle mean compared to the more usual
  ground symbol?


Conceptually, the same. All of them mean grounds (0 V reference points, for their "local" circuitries), but different ground symbols mean different grounds.