The characters in my game have excellent backstories, but bringing them to the forefront, highlighting the skills, abilities/ skills/ feats and NPC connections they made sometimes isn't the easiest.
This is not the players bringing up the connections, but me as the DM doing so, bringing up something relevant from the past into the present, and helping to "connect the dots" between past and present to make a seamless narrative.
In my experience, mining ideas from PCs' backgrounds can be a bit tricky. That's because a relationship or event written into a backstory can serve one of several quite distinct purposes. For example:
My character is a poor, pious cleric. I write down that she has an evil bishop brother because I think it would be fun to see her confront corruption close to home. Nothing would make me happier than to see my evil brother evolve into the game's big villain for a while.
We're playing a supernatural investigation game. My character, and FBI agent, has two kids. I didn't make up the kids because I want the GM to threaten them or use them for plot hooks; I just want to show that my character is a family man.
I'm playing a pirate wizard in our pirate game. We've previously determined that learning to be a wizard in this setting is an arduous and very formal process. I write down that my character was trained as a wizard but then turned to the outlaw life after being falsely accused of a crime. That detail isn't especially meaningful to me, it just explains how the pirate and wizard parts of his background fit together.
At times, trying to divine what exactly a player hopes to do with a particular bit of a backstory can be tricky. In my experience, there are two particular issues to watch out for:
In many games, a PC's background is the main way in which non-GM players create details about the world. Bringing that stuff into play is good, but it's possible a player will feel disempowered when you start putting your own spin on their relationship characters or make their backstory part of a plot twist.
Character backstories are often about defining a personality. Sometimes bringing in an element too directly means some fundamental character-defining conflict is resolved, and a player feels like their PC's arc as a protagonist has concluded prematurely.
Luckily, it's easy to avoid these missteps with a bit of communication.
How do you know which elements of a character's background their player really wants to bring up in play? Just ask.
Some games use game mechanics to guide these discussion; these kinds of mechanic are called "flags" (most clearly discussed here: practical examples of good flags, some technical details). You can add a basic version of this framework to existing games pretty easily. Here's one example of a quick-and-dirty method.
For backstories specifically, ask players to highlight what they want to see reincorporated in play.
For example, ask everyone to write down, separate from any long-form backstory:
You can look at these sheets during planning or in play to help guide your GMing. Encourage players to modify them over time.
If you're using the method above, look at these notes during planning and play. Make it a part of session planning, encounter design, or scene framing. Incorporate designated major backstory characters into your own relationship maps, side-by-side with the NPCs you've invented yourself.
I've made good experiences with using each player's backstory to create a personal quest. For example, a bard in the Princes of the Apocalypse adventure mentioned a rival that made him lose his patron. He described the rival as a fire genasi bard (the character himself was a water genasi). In this case, this fit perfectly, since there's a fire genasi called Bastian described in the adventure, located in the fire temple. If there wasn't one, you could easily place him into a suitable location.
In my games, I found a seed for a quest in every character's backstory - usually one or more NPCs, a faction or an item, but it could be anything... I usually talk to each player quickly players about the details I pick from their story for the personal quest, to make sure I got everything right and allow them to refine it a bit to their liking, and then write them a short 'quest description' in five lines or so before the first session (which they can accept, or suggest changes to).
In earlier campaigns, I gave out a mechanical reward to the player when he completed his personal quest. However, I found that this led to rivalries within the party. Now I give out the reward to all players, and just a minor extra to the character whose quest was completed. This greatly improved the interest of all players in other character's backstories. This also solves the problem that in free-form and mainly location-based campaigns, not all characters might get the bonuses, only the ones who are lucky because the party decided to go to the village/... where their personal quest was located.
Make sure the personal quest is not plot-relevant for the main story you planned. Never force a personal quest - it should always be optional. However, I made good experiences in connecting them loosely to the story. Going back to the example above, Bastian the genasi in the fire temple is a medium-rank minion of one of the evil cults the players were after.
Pick an element from their backstory, make it into a personal quest, place quest-relevant hints, items and/or NPCs in locations they are likely to visit. Hand out party-wide rewards on every completed personal quest to raise interest in each other's backstories.
