Unlike a cell phone, casting involves hardware that the Streaming Stick just doesn't have. The Roku is designed to output only one way, via hardware connection. Why not simply run the Roku website on your PC? That's what it is designed for.
The Roku Stick is a client, not a server. If you're looking to capture video from the Roku stick you'll need a video capture card or dongle that can take in HDMI, such as (but not limited to) the following:
You then connect your Roku directly to the PC and can view or capture the output.
I don't think the PC Roku app can actually play content on the PC, it just plays PC content on the Roku (PC as server, Roku as client).
If you just want to watch your TV on your PC you'd do better by going to the web sites (or apps) for the content provider concerned. They will almost always have the facility to watch your content that way. For example Netflix, BBC iPlayer, ITV Hub, All 4, Demand 5 and UKTV Player are all available easily on the PC and that covers all the main stuff I watch in the UK. The situation is much the same for other content providers.
What Andy said. Roku has no method of sending any audio/visual signal out other than HDMI (or composite on the Express +). It is a simple media streaming device, and has no ability to cast or display itself by any method other than the HDMI/composite ports.
Dan Roku Stick (3600), Ultra (4640), Premiere (3920), Insignia 720p Roku TV, Sharp 4K Roku TV, Nvidia Shield, Windows 10 Pro x64 running Serviio and Plex on a wired Gigabit network.