What should I do with my purple cone flowers now that the flower part has died? The purple petals are all gone now and the middle is turning black. Do I cut them down to the ground? Clip off the heads? They were so beautiful when they were blooming.

BDB-
The secondary flowers are rarely as nice as the primaries but good luck. If you want the plants to rebloom this year cut them back to the nodes. These are little bumps or maybe even shoots of new growth emerging, usually above healthy leaves. This will accelerate the bloom and is known as deadheading . If you allow the plants to form seeds, their energies will focus on that process.
Tony M

You can also plant new transplants grown from the seeds in the dried flowers, and if you go ahead and try to germinate them now, then transplant into the garden when it cools off a little, you can probably get more blooms from these new plants next year. They do look a little ugly with those dried flowers, don't they? Half dead and half alive.

I let the seed heads form and just leave them - ugly is relative I guess. I think they're sort of interesting looking. The birds like to eat the seeds and also some usually germinate; I have increased my planting from just three plants purchased several years ago to more than 30 now, all "volunteer". I have a nice big patch that looks gorgeous in the early summer and have had enough to share with other gardeners.