According to the C11 standard (read n1570), char can be signed or unsigned (so you actually have two flavors of C). What exactly it is is implementation specific.
Some processors and instruction set architectures or application binary interfaces favor a signed character (byte) type (e.g. because it maps nicely to some machine code instruction), other favor an unsigned one.
gcc has even some -fsigned-char or -funsigned-char option which you should almost never use (because changing it breaks some corner cases in calling conventions and ABIs) unless you recompile everything, including your C standard library.
You could use feature_test_macros(7) and <endian.h> (see endian(3)) or autoconf on Linux to detect what your system has.
In most cases, you should write portable C code, which does not depend upon those things. And you can find cross-platform libraries (e.g. glib) to help you in that.
BTW gcc -dM -E -x c /dev/null also gives __BYTE_ORDER__ etc, and if you want an unsigned 8 bit byte you should use <stdint.h> and its uint8_t (more portable and more readable). And standard limits.h defines CHAR_MIN and SCHAR_MIN and CHAR_MAX and SCHAR_MAX (you could compare them for equality to detect signed chars implementations), etc…
BTW, you should care about character encoding, but most systems today use UTF-8 everywhere. Libraries like libunistring are helpful. See also this and remember that practically speaking an Unicode character encoded in UTF-8 can span several bytes (i.e. char-s).