Meta Battle Subway PokeBase - Pokemon Q&A
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How often is it and how do you increase it?

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The more pokemon you capture the higher the chance for a critical capture

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The probability of a critical capture occuring depends on the number of Pokemon you have caught in the PokeDex

The higher the number of Pokemon caught, the higher the chance of getting a critical capture (Specifics in source).
So to increase the chance, simply cath more Pokemon.
Critical captures can fail, however they have a much lower chance of doing so, because it only makes one shake check. Normally, a Poke Ball shakes up to three times, but because when a critical capture occurs it only shakes once, it only has one chance to fail, rather than three. There are more details in the source.

Source

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How often dose it do it or is that in the link?
It's in the link, I believe the base chance for it to happen is close to the unmodified catch rate (30%), and this chance rises the more Pokemon you have registered in your PokeDex.
No; the "base chance for it to happen", if there is anything of the sort, is the unmodified catch rate divided by six, when the Pokédex modifier is 1. Note, however, that at the start of the game, before you've caught many Pokémon, the critical capture chance is zero, and the Pokédex modifier doesn't become 1 until you've caught over 150 Pokémon.