Green Screen Streaming
There are several ways of achieving this.


Use a hardware solution. They are available from Datavideo or on most video switchers. Obviously, these cost money and the results are dependant on how well you've lit your setup.   
For a software solution you will, like you've mentioned, need to connect your camera's video output to your computer. (It seems you have an HDMI output) Therefor you will need an HDMI input on your laptop. If your laptop supports USB3 or Thunderbolt you can look at solutions from Blackmagic Design (among others) to capture the HDMI feed over USB3 or Thunderbolt. The Thunderbolt solution is really affordable, the USB3 solution, less so.  
Some of the BMD models allow you to do Live-Chromakeying in the Blackmagic software.  
For Windows I came across this software, http://vidstudio.com/ it seems free and allows for live-chromakeying.
For the Apple filmmakers among us. A quick search on Live-Chroma Keying OSx will result in a large selection of affordable software.  


Especially when you are going to do the actual keying in the post process, you don't need the highest quality live killer. These are often times very expensive and give moderate results at best.