I've never used the API myself so I can't really show any example. And making an example would mean a lot of reading and trying etc... so I'd put that task on you.
Scour the manual for AJAX construct.net/en/make-games/manuals/construct-3/plugin-reference/ajax
The rough outline will be
1. Send an AJAX GET (request URL) request to the API with the correct keys/userid etc.
2. Load the response into a JSON object (parse AJAX.lastData)
3. Validate it. If it only contains "errors": ["invalid download key"] or similar you know the user does not own the game
There's also this itch.io/api/1/KEY/game/GAME_ID/purchases which probably is useful for your case since it requests to check a users purchases and you can verify it includes your game/dlc/extra download.
Note that you have to replace certain parts of the API urls with your data
itch.io/api/1/KEY/game/GAME_ID/purchases
KEY = API key you can generate somewhere in your profile
GAME_ID is the game id you can get from a different API call (or maybe somewhere from your page too? Not sure)
Just try around and if you cannot figure something out you can always come ask again. But I do recommend banging your head against it at least a little bit, it's a good learning experience.