First, you need to compile the C library. Do something like:
gcc -shared example.c -o libexample.so
Of course, for a complex existing library compilation could be considerably more complex -- if you're wrapping an existing library, it probably comes with some sort of Makefile to help you build it.
Then, in Lisp, use CFFI to define and load the library. This seems to be the main part that you're missing.
(cffi:define-foreign-library libexample
(t (:default "libexample"))) ;; note no .so suffix here
(cffi:use-foreign-library libexample)
This part:
(t (:default "libexample"))
is a conditional which you can use to give different loading instructions for different platforms. (t ...) is the catchall option, much like with COND. You can find the exact syntax in the documentation for define-foreign-library.
You would now normally use cffi:defcfun and so on to define the functions in the library. This is what the SWIG-generated file does for you, so load it:
(load "example.lisp")
You can now call the functions as normal Lisp functions:
(fact 5)
=> 120