What (or how) can I insert a box with a visual of my face while I am recording the game I'm playing? I currently use Hypercam but there is no option for this game/reality split screen.
Xsplit is a popular software choice for people recording and/or streaming video games. It allows multiple inputs and can overlay video on top of video, as you'd need for the PiP effect. There is a small fee to purchase this, but would highly recommend it if you are serious about it.
Another popular solution is Open Broadcaster Software (OBS) which is similar to Xsplit, but is completely free. Here is a guide to using OBS. When you setup a "Streaming Profile," you can just set it to locally record.
With both services, you have the option to record footage locally (such as to edit/produce and then upload to YouTube) and/or stream directly to an online streaming service (such as Twitch or Ustream).
This is normally done as a post production operation. Record the video of you and the gameplay and then sync them up in any video editing or composition software and it should easily allow you to scale down and position the clip of you over the video game layer.
It will normally either be called picture in picture on more consumer targeted software or is done by altering the scale and position of a video layer in professional products.