Pokemon-go – What are the chances of finding a Shiny Magikarp

Honestly I am simply translating your comment section into an answer and adding a bit. The algorithm is random (though I have no source) when it comes to Pokestops. Some times you will get 2 Pokeballs and 1 Potion, and other times you will get 1 Pokeball, 1 Greatball, an Egg, and a Potion. There is an unverified rumor going around [@Kevin L] about hitting your 10th unique Pokestop. They say every 10th unique Pokestop you spin you will receive a bonus - 50 extra experience and multiple extra items (still random). I hope this helps sum things up.

This is entirely random. There is no way to trigger a Critical Catch, and the chance isn't based on anything to do with your progress in the game.

Source 1 (Emphasis mine):

Critical catches happen completely randomly, and it's triggered by the game, so if you don't see the effects described in other posts, it's not critical.

Source 2 (emphasis mine):

Critical Catch is a mechanic in Pokemon Go. It is a random occurrence that guarantees that the Pokemon is caught

As to what exactly this chance is, that information has not been released, but as far as I can tell from all my experience in-game, and online research, this is entirely random.

The only other piece of information I have found about this is the fact that the "critical capture" is entirely client side, and doesn't actually mean anything, other than a "faster animation"

System efficiency requires that as much work as possible is delegated to the client, and that there is minimal data transfer. Therefore it's likely that the server only decides and reports whether the catch was successful, leaving the client to decide how to animate that result.

Source