Swimsanity!

I'm not yet sure what unlocks new characters, but since no guide exists, I thought I would offer some speculation. I'm guessing it's gold orbs. Though I was mostly unaware when getting new characters, I unlocked my final character after surpassing 150 gold orbs. I "earned" many of them them by completing challenges in versus modes locally with an extra controller. The list of challenges and your progress can be found by selecting the "Gold Orbs" box in the upper-right of the screen, below your username.

After having all characters, it's as simple as playing once with each and using its unleash ability.

Characters: Harpoon, Shock, Medic, Piranha, Mooba Mech, Babyfish, Harpoon Drones, Bomb Drone, Slow Mo, Hyper, Aqua Tank, Vortex, Shark

Edit: Thanks to BluesWolf66 for catching my omission. I've added Shark to the list of characters. I created the list from the character selection in adventure mode, which doesn't include Shark. It is, however, available when selecting characters in versus modes. I hadn't noticed this before, so this is worth noting. Just be sure to use versus to perform your Shark unleash.

Edit 2: I spoke with a dev, ILL Khalil, who told me that character Babyfish is unlocked by beating all adventure levels. I have Babyfish, but haven't beaten Piranha Bay. Please comment if you can confirm.