How can I use C API for PYNQ Z1 board programming?
PYNQ provides Python APIs.
If you want to use C, PYNQ does not provide C APIs. You can build C applications like you would for other Zynq Linux boards. You would be better posting on Xilinx forums or elsewhere if this is what you want to do.
You can call C code from Python which would allow you to work with PYNQ. More general info on this here: https://realpython.com/python-bindings-overview/