Bonds and Rivalries
When you kick off a new campaign and you and your fellow players start designing your new characters, it's easy for them to all wind up fairly autonomous. After all, unless you and a fellow player decide your characters know each other, you just aren't likely to factor the other player characters into your backstory. And if the campaign's storyline kicks off by throwing your characters together as strangers, then that's not an issue.
However, if your campaign opens with the words, "You've all been working together as a group of adventurers," then you're going to want to establish some connections between characters.
There are several different exercises out there to encourage player characters to form connections, but unless you're playing a game like "Fiasco," most of them aren't baked into the DNA of the game itself. In those instances, the responsibility falls to the GM and the players to establish those inter-character connections.
My friend Lyn recommended a great version of this process: When creating your characters together in your Session Zero, you each give a basic description of your character concept. Then, each player shares with the group a story about how the character to their left has pissed them off, and how the character to their right has saved their lives in a previous battle. This means you're forced to make up details and traits about the other characters, and infer characteristics from the shared stories. Then, everyone actually writes up their character sheets, and incorporates in those aspects they discussed together.
At the beginning of my first Tyranny of Dragons campaign in 2013 (which, as many of my campaigns do, began with the characters thrown together by circumstance), I prompted a few members of my group to form connections with each other. We had two characters who were more scholastic in nature (the monk and the warlock), so we decided that they had been traveling together, and I asked them to decide how they felt about each other. I also asked our bard charlatan if I could have him already be under arrest by our paladin. He was game for it, because he knew it would foster an interesting dynamic between the two of them (and indeed, it really, really did).
Now, it's not just necessary to form these connections within the group; it's just as important to build them into your character's backstory as well, so they have bonds and rivalries with other NPCs. That's why 5th Edition D&D added the sections for "bonds, flaws, and ideals," and it's why Star Wars: Edge of the Empire has "Obligation," where your character is beholden to a flaw or dilemma. Your Obligation might be an addiction, a favor, a large debt, a family member you are concerned about, an obsession, or a bitter rival.
And Obligation isn't just flavor - at the start of every session, the GM rolls to see if your Obligation is triggered. If it is, everyone suffers strain damage, because your character can't get their mind off their Obligation. There's also a pretty good chance that triggering your Obligation will cause it to play a role in the story itself - a bounty hunter might show up looking for you, your little brother might get into a tight spot that he needs you to bail him out of, or you might be actively tempted by your addiction (or run out of whatever it is that you're addicted to).
In between sessions of my campaigns, it's extremely common for me to text some or all of the players and ask for more details about their past. I usually already have a few aspects of each of their backstories, but I want the players them to consider their pasts in more detail. For example, I don't just want to know who our barbarian had loved and lost, but how they had been separated, and when they had last seen each other. For our warlock, I want to know what she knows about her otherworldly patron (how he appeared to her, what the source of her power is), and what she doesn't know (his intention, his alignment, etc).
These questions, however, are only important if they matter to the story. Sure, it's helpful for your characters to come up with some of their backstories, but unless they somehow factor into the story you're telling, they're just an interesting bit of trivia. So, start finding ways to use those bits of trivia as pressure points.
The aforementioned bard charlatan's whole gimmick is that he goes to new towns in new disguises, performs as a new musical act, and then casts "Sleep" on the crowd and robs them all blind. So, when the story took our players into the city of Baldur's Gate, it made sense to add in an NPC who remembered the bard, and wanted to get revenge. And not only was it a chance to show the character the consequences of his actions, it shined a light on his behavior for the rest of the party to see.
This article isn't just for GMs, though; it's for players, as well. If your GM doesn't necessarily urge you to form bonds between Player Characters, that doesn't mean you can't or shouldn't still do it. The bond between the monk and the warlock in our Tyranny of Dragons story came about because their players were discussing their characters, and it just made sense that they'd be traveling together; I had very little to do with it. If you decide it makes sense for your characters to have a connection, then by all means, go for it!