Recently, I was asked a similar question, if I could give a list of most Squiddles. That post can be found here: (x)
However, you've asked for backstory. What I'm going to do is give you backstory on five main characters, Plumthroat, Princess Berryboo, Creamsicle, Skipper Plumbthroat and the Elder Squid. I apologise, but it just takes far too long to give a detailed analysis of every character. If you'd like to look at the wikia, there's a link on the main blog.
That being said, let's get to it. Long text post coming up, spoilers abound.
Plumthroat, or, P - Plumthroat's mother is suspected to have been close with one of the few human characters, Skipper Plumbthroat. It's said she grew up with him. However, she died giving birth to P (a fact which was only briefly spoken about in one episode), having only enough time to name her Squiddle son after her friend before passing away. P grew up believing that his mother's death was his fault, and he's very angry, guilty and sad because of this, although he hides it well.
Princess Berryboo - Berryboo, being the daughter of the Queen of the Squiddles, grew up receiving anything she wanted. She stayed in the castle all the time, trying on her jewelry, playing with her expensive toys, forcing the servants to play with her although they hated it. She was very spoiled, snobby and bratty (and still is at times), causing most of the castle to develop a deep disdain for her. She was very sad because of this. One day, she swam away from the castle and met Plumthroat. Although she was rude to him, he introduced her to the rest of his friends. With their help, she discovered what friendship truly is, although she's still hitting some snags along the way.
Creamsicle - We don't know much about Creamsicle's backstory, despite her being a main character. What we were told though is that she is Plumthroat's lost sister (found in season 3), separated from him at birth and sent to live with her father.
Skipper Plumbthroat - Piecing together small, fleeting facts spread throughout seasons, it is discovered that Skipper Plumbthroat used to be friends (perhaps more) with Plumthroat's mother before she had children. It is suspected that he was in love with her, but could never say, as a human/Squiddle relationship would have been condemned. When she died giving birth to Creamsicle and Plumthroat, Skipper developed a deep hatred for them, despising them for taking his friend/love from him. That hatred is still fueled today.
Elder Squid (also known as Grandpa Squid) - the Elder Squid is a very mysterious character. All that is canonically known about him is that he is Berryboo's maternal grandfather. He is a community leader and is always there to help the Tangle Buddy group when they need it. He is arch rivals with Skipper Plumbthroat, something which seems to stem from childhood (going off of a quote from the sixth season). It is rumoured that both he and Skipper had a rivalry over Plumthroat's mother, a rivalry which is still going.
Phew! That was very, very long, even with only five characters. I hope that's enough for you, Anon, and if you ever want another one, just send me an ask. I might make a page on the blog that has information about each Squiddle, since I get asked about them quite a bit (not a bad thing)!
Sorry for clogging up anyone's dash! Photos are taken from the Squiddle wikia.