Based on the Arduino sources from https://github.com/arduino/Arduino
You have to compile your code with avr-g++ (NOT avr-gcc) as the Arduino sources contain c++ classes that cannot be used in c without adaption!
In your project you have to define the preprocessor variable BOARD either in your code or with the compiler flag -D.
See local/config.h for valid values of BOARD.
In your code first include the config.h, then the Arduino.h:
#include "/Path/to/ArduinoLibrary/local/config.h"
#include "/Path/to/ArduinoLibrary/src/cores/arduino/Arduino.h"
For compilation use the following compiler flags:
C: -Os -Wall -fno-threadsafe-statics -fno-exceptions -ffunction-sections -fdata-sections -mmcu=$CPU
C++: -Os -Wall -fno-threadsafe-statics -fno-exceptions -ffunction-sections -fdata-sections -mmcu=$CPU
Replace $CPU with the AVR chip on your arduino board (e.g. atmega32u4 on the ArduinoMicro).
In the linking process include the Library with the following Options:
-L/Path/To/Library/File -larduinolibrary