ADTPro is what you want. Visit here :
http://adtpro.sourceforge.net/
The ADTPro software allows you to bootstrap an Apple II from the 'bare metal', with no existing bootable disks. You just need to make up a serial cable to connect the Apple IIc's 5 pin serial port to the 9 pin DSUB on your adapter. Don't cut up the adapter, just use a mating DB-9 connector (available at Radio Shack). Rat Shack used to carry 5 pin DIN plugs as well, but I don't know about now. Shouldn't be too hard to find an AT keyboard extension cable, or a MIDI cable, or something similar to cut up. IIRC, Radio Shack still sells MIDI cables, so you can buy one, cut an end off, and solder on a 9 pin connector. Anyone remember if MIDI cables have all 5 pins wired through?
ADTPro is fall over easy to use, and once you've got the necessary serial wiring set up, you're in business. All the documentation, including wiring diagrams, is right on the ADTPro site.
You'll want to check your IIc though, see what ROM version you have. If it's a real early one, you might have trouble running the serial port at high speeds. To check which version you have, enter the system monitor and type FBBF. It'll respond with one of a few things:
FBBF-FF - Original IIc. May have flakey serial ports
FBBF-00 - Updated IIc, supports UniDisk 3.5
FBBF-03 - Expandable Memory version of IIc
FBBF-04 - Last ROM revision, minor bug fixes
-Ian