The screen mirroring that roku provides is 'miracast' which requires a wifi-direct connection between the devices, and not all PC hardware can do that. Chromecast works over your existing network connection but the screen mirroring part does not work on the roku. Some applications provide their own 'casting' function where the phone app tells the TV app what content to play but the TV app gets its own copy. YouTube/Netflix/Plex and possibly others may work this way on the roku (you'll know because the phone app itself will show the cast icon).
Generally, screen mirroring of some other device is inefficient and not the best way to view things even where it does work. If you are playing content from the internet, chances are there is a roku app to play it directly. If you are playing from files, you can use a server like Plex on your computer to send them, or put them on a USB drive to play with the roku media player, or put them on a phone or tablet and send them with the roku app.
If you really, really need chromecast protocol, get an android TV instead of a roku or a separate chromecast device. Or, if you want a perfect display from your computer screen use an HDMI cable to the TV.