I've been involved in several conversations of late where other GMs and players have lamented that their groups back home don't have the kind of games that the groups I'm in. In specific they want the interpersonal relationships and strong character growth that the groups I'm in have managed to get. Often when talking about the issue it boils down to their game is more event based - and people don't go for that depth of RP. I'm not going to lie, willingness to participate is part of it. Your group members need to trust the group as a whole. However, the other thing is easy and can be thrown into whatever game you're running. Today, let's talk about that.
Challenge Is The Key To Development
First off, if you want to see character growth and development than you need to do one thing: you need to challenge your characters. Don't worry, this isn't as hard as it sounds at first. Most character sheets will tell you how to challenge the character. See those disadvantages? Ping them, but don't just ping them for mechanics, ping them for social reasons.
Does a character have an Addiction? Take away their access to their drug of choice. A dependent? Make stuff happen that the dependent becomes a social issue. Are they unlucky? Have random bad things happen.
You can also just ping them in areas where they're not strong. Put the combat monster in a social situation. Put the face in a fight. Don't let the face help the combat monster, or vice versa. Deny the characters items they want - not by saying no to them, but by making the items 'unavailable' in game because they belong to someone else. See what happens.
There's a thought that basically states you can know someone for thirty years, but the day you put them over a fire is the day you're going to learn who they are. In other words, we find out who we are when we're put to the test. You want to know who the character's in your game are? Put them to the test.
Putting PCs to the test frequently brings back one answer: the PCs are horrible people. Expect to find out that some of the characters in your game live up to the tag of murder hobo. Also that they may not be the nicest people when put to the test. It's not a bad thing, it's just a thing. If anything, it's part of the escapism for much the same reason that people play Grand Theft Auto.
Again: don't fear your PCs being murder hobos or "bad" people. Just expect it. Heroes don't have to be good people.
Now We Know Who They Are
So we know who the characters are. We know how they want to respond to situations. So now what do we do?
Well, we see what they do when they can't do the thing they want to. In other words: we poke the bear.
This is basically another way of saying focus on what the PCs will do, instead of what they can do.
For example, in one of my games I had PCs that liked solving problems with full frontal assault. I don't blame them. They were super heroes and that's kind of how hero games go. However, I wanted to poke them that violence wasn't always the answer and so I made a villain that they couldn't beat with the combat mechanics in the game. They fought. It was epic. People got hurt. In the end the PCs couldn't take down the villain and they learned a lesson: they weren't strong enough to take that guy down.
Not the lesson I wanted them to learn, but still a lesson. There were several more encounters with this villain. He even killed one of them because the character wouldn't back off even when already drastically hurt. However, the PCs did change their approach. They were more cautious going forward. Took more care with future unknown villains. In the end, they still changed and grew.
Consequences Shape Growth
People are relatively simple creatures in a lot of way. We tend to develop down the paths of least resistance, based on our personal likes and what not. We learn and develop tendencies to not do things based on resistance. Sometimes this resistance is pain - like how we learn to not touch fire. Others are taught other ways.
Get punished for lying? You probably don't lie very often. The consequences of stealing are meant to prevent theft. The same for other things in life.
Now, obviously in RPGs you can't punish the same things we punish in real life. However, there are things we can do. Do your PCs kill people too fast with little provocation? Negative consequences could teach them this isn't always a good thing. Furthermore, positive consequences for trying to talk things through first could cement the fact that other paths are open to them.
Remember that last part by the way. You can't teach with just negative consequences. You have to provide positive ones as well.
Don't Try To Mold The PCs
This may sound strange considering this post is how to make a PC grow and change, but don't try to mold the PC yourself. Your job as the GM isn't to shape the PC into what you want, but rather to poke the PC and make it change naturally. Things like resistance to being a murder hobo are fairly natural, but if the PC starts to develop in a way that is just more of a hidden murder hobo don't judge the player or character harshly because of it. Play the world, poke the character on occasion for development and discovery, but don't try to force it in a certain direction.
But It's An Event Based Plot
That's great. Believe it or not, this isn't a full time job. Your event based plot can still happen, and in fact makes a great way to hide what you're doing here. Just take a few chances to poke your characters. It doesn't have to be a lot of development for an Event Story, but you can still have some.
Poke the characters. Put them over the fire. See who they are, and then see what happens when you challenge who they are. You might be surprised how much it can enrich your game.