I would like to re-program in C the program Burp (https://portswigger.net/burp), at least the part with the proxy.
Burp starts on the computer a proxy over which then the clients in the network "log in" or "access" can. A CA certificate from Burp will be installed on the clients. From now on Burp can also read HTTPS traffic.
I would like to re-program this principle in C.
I do not know which libraries to use for - the proxy - decrypting the data with de certificate key
Many people already wrote a proxy in C (for example, nginx).
You can often look through their C code and discover what they did and which libraries they used.
As for the network layer, I am biased since I'm the author of facil.io, which I love... But a quick search will offer libev, libuv and libevent as very common choices that support more platforms (such as Windows).