c++ library with c interface
You can use the PIMPL idiom in the C wrapper. You provide a method YourClass_Create that internally calls the constructor (using new) and returns the pointer to your class instance; for the client code this will be just an opaque handle (it may be a typedef for void *), to be passed to every function of your C interface to specify on which instance it has to work (just like FILE * in stdio).
All these functions will have to do is to call the corresponding method on the handle (converted back to a pointer to your class) and translate exceptions to error codes.
As @jdv-Jan de Vaan pointed out in his comment, don't forget the necessary #ifdefed extern "C" {} around your C wrapper code, otherwise you may get linker errors.