There are two public accessible points for lake Eyre that I know of.
1. Level Post Bay. This is approached from Maree via the PAR on Mulorinna Station. You get to drive the country between Lake Eyre South and Lake Eyre North and see the Goyder Channel that joins them. There is no camping at Level Post Bay, but there is a campsite at Mulorinna with donation toilet last time I was there. From memory, it's an hour or two from the Campsite to the lake itself.
2. Halligan Bay/ABC Bay. This is the most well used track that runs from the Oodnadatta track a few kms south of William Creek. The track condition is variable and if there has been much traffic (yes, there has) then expect a badly corrugated sandy (goat) track. Just take it easy and you'll be fine. While people have driven out there in cars, you really need a 4wd, and something with good clearance is the way to go. There is good outback camping at both Halligan Bay and ABC Bay. Toilets at Halligan Bay but better lake access at ABC.
Br aware of the arid conditions - take lots of water! Halligan Bay track is where the
Austrian tourist died trying to walk back to the Oodnadatta track after bogging their hire 4wd. Also, make sure you understand your vehicle and can recover it in case it becomes bogged - tyre pressures, winch, maxtrax, whatever. Usual emergency precautions apply: PLB/Spot/Satphone/HF Radio - take
at least one of these!
You can view Lake Eyre South from the Oodnadatta track near Curdimurka. There is an unofficial access to the south lake near there, but there is a barney going on and I believe it has been closed off.
Hope that helps!
Michael