Don't link them, at least not without some kind of isolation device.
You risk grounding issues, at least.
If you run a wire from pin 16 of the ATX connector (usually a green wire), to pin 17 (or one of the other ground pins/wires, usually black wire), the PSU will turn on when mains power is applied (from the wall).
If the PSU has a mains switch, you're done.
If the PSU doesn't have a mains switch, run a wire from pin 16 to a toggle switch on the chassis, and then back to pin 17. You can then turn the drives on and off with that toggle switch.
Put the PSU in the enclosure and hook it to the drives, and turn it on and off as needed, like any other external drive enclosure.
Another option would be to use only one PSU, physically bolt the two chassis together, cut access holes in between, and run (perhaps extended) power and data cables as needed.
If you really want to link them, then:
- Get a SPST NO (single pole single throw, normally open) relay that's triggered by 5V, preferably of the photo-isolated variety (example). At 5V you could probably also get away with an opto-coupler, or even a transistor.
- Tap the Power Good pin and a ground pin on the computer's ATX connector, run wires from those to the trigger poles of the relay.
- Run wires from the Power On pin and a ground pin of the enclosure's PSU to the load-side poles of the relay.
When the Power Good line in the computer gets signal it will close the relay, causing the Power On in the enclosure's PSU to be grounded, turning on the PSU. Same in reverse Power Good goes away, Power On gets disconnected...