Android Connected to Multiple WiFi I recently got a new smart tv and a samsung s6. I have an ambitious project I would like to pursue. I want to do some screen mirroring, this works fine but with my wifi being fairly slow any real streaming lags.

I would like to set up a private home network which would connect my phone and tv, this network would be free from any other traffic. I would then require my phone connect to my incoming wifi network.

I want my phone to get media from an outside network and stream it to my tv over my private network. Is this possible? Where can I even start looking?
The solution you want is not possible because of both hardware and software limitations. On the hardware side you need a device with 2 WiFi antennas which neither phone has, and on the software side, I never heard that Android supports multiple WiFi connections or antennas. 

Another thing is that Screen mirroring is using WiFi Direct, meaning it connects the TV with your phone directly, not over a router or access point, so your home WiFi shouldn't have any impact on the mirroring itself.