How was the cake poisoned at the Purple Wedding?

In the show, it is not explicitly explained how Joffrey was poisoned. The medium of poison is unclear, whether it was the wine or the cake.

However, in S04E06, Grand Maester Pycell presents Sansa's necklace as a proof in the trial against Tyrion. And since Sansa was wearing the necklace throughout the wedding and nobody but Olenna Tyrell goes up near her for a small talk, there would have been no time to mix the poison in the pigeon pie (which you confused as cake). Hence, the poison was certainly mixed in the wine.

Littlefinger & Olenna were directly involved. Olenna even confessed to it in front of Margery in S04E04. Ser Dontos was hired to transfer the poison onto the unsuspecting Sansa.

If you go back to S04E03, you can clearly see the point at which, Olenna takes the poison from Sansa's necklace & puts it in Joffrey's wine.

If you don't want to rewatch the episode, here is a detailed explanation with screenshots from that episode thanks to r/zephytr

The cake was nothing more than dry and full of pigeon guts, after what Joffrey did to it :)

Joffrey was poisoned by The Strangler, a poison that is formed into a crystal through a deeply intricate process- A Clash of Kings gives a fairly detailed history and description of how The Strangler is created. Remember at the beginning of Season 2, when Maester Cressen attempted to poison Melisandre by slipping poison into her wine? He, too, attempted to poison her with The Strangler. The Strangler is not "activated", so to speak, until its crystal was placed in a liquid and dissolved. Maester Cressen needed to slip his Strangler crystal into Melisandre's wine, and Olenna needed to somehow get her Strangler crystal into Joffrey's cup.

Olenna's Strangler crystal came from Sansa's necklace (in the books, I believe it was a hairnet) that Dontos gave her. Although it is well hidden on the show, and not something you'd likely pick up on your first watch (assuming you didn't know what was coming), Olenna plucks one of the crystals from Sansa's necklace at one point during the feast. Olenna must have slipped the crystal into Joffrey's cup between then and when the cake was brought out.