One of the things that made these comics interesting to me was the fact that they focused as much on the characters of the books as they did super-powers, sometimes the focus was even more on the characters.
This is good because on of the things that tabletop RPGs do well is to focus on the player characters and their interactions. For many gamers, whether with new or old school approaches to gaming, this is why they game...to tell stories around their characters. For fans of these kinds of games, it makes comics like these excellent models for their games/campaigns.
One of my longest running Classic Marvel Super-Heroes RPG campaigns was influenced by these two books, mostly because they were what I was reading each month as I GMed the campaign. Roleplaying was important to these campaigns, and while we would have elaborate knock-down, drag out fights in the games the players also spent a lot of time talking and developing their characters. Relationships happened. Characters married NPCs. Characters died as players left the group, or decided they wanted new characters. It was interesting because, when we started the campaign, most of the people had never played the game, so I asked them what they wanted for a character and modeled it for them in the rules, or game them a character from my notebooks and they ran with it. One of the cornerstone characters of our campaign, a stereotypical conservative super-hero named Real American, was based off of the character of Golden Boy from the Wildcards novels. The player (who was not all that conservative in real life came up with someone who was a play on conservative super-heroes like Guy Gardener) took the bones of the character and molded a new personality and motivations for the character and made it his own.
However, this post isn't about how to appropriate characters and tropes from comic books and to use them in your super-hero games. That could probably be a post all on its own.
Honestly, you have to have the "right group" of players if you want a game that is going to focus on characterization and interpersonal relationships. Not wanting to do this isn't a bad thing, but it isn't going to be what ever group is interested in doing (or even capable of doing). You have to be upfont about wanting to run this sort of game, so that players do not have the expectations that this campaign will be more "standard." There is a certain type of player who wants to fight everything all the time, and while they may have a place in some games...in others they can be a detriment.
Now, obviously, you can play this sort of game with any type of roleplaying game, if that is what you want to do. We did it for years with the Marvel Super-Heroes game, so it can be done. Again, though, not everyone is going to want to use a game and "never touch the dice for sessions." They are going to want some sort of mechanical basis for these sorts of interactions. For that, I suggest going with their preferred game to handle these sorts of things mechanically.
One of the first things that you have to come up with, for this kind of game, is a strong theme. For the Teen Titans you could say that the theme is "Teens coping with their powers and difficulties by joining together and helping each other." For the Suicide Squad it could be "Misfits and criminals looking for redemption." The theme for the Justice League of the time could be "B-List Heroes Looking For Recognition." You could probably come up with a couple of variants on these themes, or different ones all together, from each of these comic's stories. The idea is to figure out which sort of story that you want to tell.
Next, once the theme is decided, the players need to decide how they want their characters to fit into this theme, and what sorts of personalities that they want for their characters. Using Fate Accelerated was our guide, we can come up with aspects for Suicide Squad stalwart character Deadshot like this:
High Concept: He Never Misses His Shot...
Trouble: ...Except When He Loves A Woman
I don't think you're often going to see two interrelated aspects like this very often in a Fate character. It fits for the characters, at least as how it was interpreted back in the 80s, and they both work. I like how they sound like a tagline from a movie poster. Some GMs might want you to make these into one aspect, but I think that would be too specific of an aspect, personally. You could change the trouble aspect into something more social like "...Except When He Wants To Fit In" instead. I like the idea of the tough as nails character who knows that he has that flaw when it comes to women/relationships. It makes for a very noirish type of character. Can that trouble be flipped to "...Except When He Loves A Man"? Of course! Play your game how you want to play it.
Use one or two of the aspects remaining to talk about the character's powers, and then fill out the rest of the character's personality. With our de-emphasis on powers, we don't need to sweat a detailed writeup of what Deadshot can do. We already know that he "Never Misses His Shot.." I would use an aspect like "His Battle Armor Is His Weapon To Kill And To Keep People At A Distance." This should be easy to invoke when a combat situation does come up, and it can easily be compelled during other times. While cliched, Deadshot is certainly an archetype for the "Checkered Past" aspect, and "Can't Take The Shot Against Batman" could round out his aspects.
I enjoy this sort of a game, but it isn't going to be for everyone. The important thing to remember when adapting your favorite comic stories to gaming is to look deeper than the surface of the stories that you are enjoying. While the flashy powers are there, and available, in the games, they don't always have to be the focus of your game. There are some really good games that are all about building and using powers. However, this is why variety in available games and playstyles is important to gaming. Ultimately what is important is that each and every group find the system and approach to gaming that works best for them and gets their game on.