For my radiator fan I've set up a circuit that requires the use of a relay
that has a switched ground on one of the pins.
The 12v wire going to the relay is powered constantly as it is connected
directly to the car's battery.
How can I make the ground for the relay only be connected when the
car is turned on?

 

Use another relay to switch it to ground, or simply reverse the coil connections so it is a switched 12V and a constant ground.

http://www.aaroncake.net/rx-7/efaninstall.htm

 

If I use a 2nd relay do I connect an ignition switched wire to it so that the ground is switched on when the ignition switched wire has current flowing through it?
Which points would I connect the wires to on the 2nd relay?

 

On the 2nd relay, connect one side of the coil to ground. The other side to an IGN switched line. Then one contact to ground, and the other contact to the first relay.

Or just use the diagram on my website.

 

Is there any advantage to having it set up this way with 2 relays over having just
1 relay?

 

Not really. Now in theory, if you use two relays in parallel to switch the fan, you have redundancy if one relay fails. But really, how often does this happen? Almost never.

I think you may be overcomplicating the fan circuit. Take a quick look at the diagrams on my website.