While I agree with Lt Dan Zsu about the Intel Integrated GPU being enough, so are the AMD APUs. Whichever is more budget friendly is the way you should go.
Also, I understand the desire to keep things small, but most slim-line/Mini-ITX cases come with proprietary power supplies that may or may not be of quality build or sufficient power. Additionally, these smaller cases generally do not allow for all the expansion slots/drive bays, we want. My advice is to sacrifice case size for a quality ATX power supply, better airflow and additional drive bays.
Finally, if at all possible, avoid WiFi for video streaming.