If I'm understanding correctly, this is something that won't involve Premiere at all (at least not until you're ready to edit what you captured).
For your screen capture, you're going to want a screen capture program which allows positioning of sources. The go-to software for this (in my opinion at least) is OBS. A lot of YouTubers and streamers use it and it's free to download and use.
If you're looking for how to use OBS, there are resources and forums on that site.
Just a little note as far as Premiere goes. FLV I think is the default recording format for OBS (because it is less likely to corrupt if the recording is interrupted), but Premiere and Media Encoder won't read the FLV format it records in, so you would need to use a third program to convert the FLV file, or set OBS to record in a different format supported by Premiere.