If you are using a PS4, you can use the internal streaming service for PS4 to record your runs to twitch, and then submit that way.Here's a rundown: http://www.ign.com/wikis/playstation-4/How_to_Use_Twitch_with_PS4

The alternative would be to purchase a HDMI capture card for your computer, and then route the video signal from the PS4 to your computer.

^ If you do this, you should export it to YouTube as YT has a better player, although not required.

And to answer your question, you need to upload screenshots to external sites like imgur. But if possible you should rather submit a video than a screenshot as a video provides more proof than a screenshot.