Get a split timer for your PC, like livesplit. Also, a way to record your gameplay will probably be necessary for necessary for verification. If you see it in the rules, you can either get a capture card setup or use an external camera to record (make sure the gameplay is visible though, best way to do this is to make the room darker). Once you have your setup, you can look at some videos on youtube on how to find games to run and how to actually go about it. Good luck!

I believe you can stream directly to twitch from a PS4. Splits aren't necessary for running, although are nice to have. You could time your run with any timer (your phone eg) and re-time more accurately before submitting.

A third option is PS4 remote play which will use your internet connection to allow the game to be played on your PC so you can use something like OBS to stream or record. This method might be prone to some delay or lag depending on your internet speeds (it used to have problems with random crashes too).

Remote play is risky option. Also, a friendly warning, be sure to enable the option to save your Twitch streams beforehand in settings, otherwise they will be gone immediately after you are done. Then highlight the runs on your Twitch account, then export the highlight to YouTube if you wish to do so. This allows you to share content on both platforms and also gives you two video links as more proof to your runs being legit.