First off, I'm going to say that there are two main approaches to storytelling.
The plot-based approach emphasizes action -- i.e., what the characters do.
The character-based approach emphasizes characterization -- i.e., who the characters are.
Both of these approaches can be used for strong and effective storytelling and characterization. The major difference is in how you learn about the characters. In plot-based stories, you see what the characters do and on that basis divine who they are; in character-based stories, you see who the characters are and, on that basis, react emotionally to what they do.
You can see the contrast between these approaches in a lot of places. In comics, the Golden and Silver Ages were plot-based; the inter-crisis stories were character-based. TV made a similar transition from plot-heavy to character-heavy. DRAGNET is plot-based; HILL STREET BLUES is character-based. LAW & ORDER, the original and still the greatest, is a plot-based throwback. LOST is character-based; VERONICA MARS is more plot-based, but is still very strong in characterization. (In fanfic, I'd say that I tend to be a plot-based kinda guy, while thete1 takes a very character-based approach.)
I think you can really see the effect of the two approaches in the contrast between Batman and Superman. In the Golden Age, Superman is KING. Silver Age, too. (Though I personally go even more for Captain Marvel.) But in the inter-crisis period, which is more character-based, it's Batman all the way.
One reason is that Batman has a much better (and better-placed) supporting cast. Most of Batman's supporting cast -- Alfred, Dick, Leslie, Babs -- historically knows his secret identity, while most of Superman's -- Perry White, Jimmy Olsen, and until the Byrne revamp both Lana Lang and Lois Lane -- historically does not. Batman's supporting cast thus can be employed effectively in more facets of the hero's life, because they know him much better than Superman's does. This gives the Batman family a distinct advantage in character-based stories; Batman's supporting cast has a wide variety of personalities and motivations, which can be brought to bear on pretty much every circumstance of Batman's life. (In the Golden Age, this didn't matter much, because the stories weren't told in a way to run with that. Which meant that Batman suffered in comparison with better-plotted heroes Superman and Captain Marvel.)
By contrast, in the Golden and Silver Ages, the characters closest to Superman, who knew the most about his dual life, were either physically removed (Supergirl, in her orphanage; the Kandorians, in their bottle; Krypto, out in space half the time; Lori Lemaris, in Atlantis) or dead (the Kents, at that point in continuity). In either case, they did not associate with Clark Kent. Superman was the social entity, and none of his supporting cast could keep up with him when he went to fight crime (though Lois and Jimmy always tried).
This is why Superman's golden-age and silver-age stories are better and more satisfying than Batman's.
It's also why Batman's inter-crisis stories are better and more satisfying than Superman's.
So, what does this have to do writing fanfic? A lot, actually.
I'm a story junkie, and I still use the character-based approach a lot. For two reasons. The first is that it's very helpful in coming up with story ideas. The second is that you can use it to dramatize these ideas.
So tip #1 for fanfic writers is: USE YOUR SUPPORTING CAST.
We do that anyway, of course. One reason we write fanfic is to show love for characters who've tended to get short shrift -- sharpest_rose, I'm looking at you! But you can use them in specific ways to good effect.
For me, the supporting cast is first and foremost a story generator. My story ideas frequently originate with me realizing that members of the supporting cast have something in common. "New Year's Kiss" came from my realization that Alfred, a former soldier and spy, probably killed people. Which gave him an avenue to understand Cassandra Cain. (This also creates mammoth potential for a team-up with Jim Gordon, also a military veteran; one of my long back-burnered ideas is that Alfred's and Jim's paths crossed briefly in Vietnam.) "Jason and Me" stemmed in part from my realization that Jason and Steph's family backgrounds had parallels: drug-using mother, father a criminal who was at first absent, later murdered.
Ask: what characters have something in common? Then ask: what exactly do they share? Then ask: how are they absolutely different? Or: how do these characters react to the idea that they have something in common? Or: what do they do about it?
Another way to use your supporting cast effectively is to have them dramatize the internal conversation of your main characters. This can be especially effective when your main character is someone who doesn't open up a lot -- for example, Batman -- or someone in a position of authority who could look wishy-washy if he tries to talk through the options himself, but looks pretty decisive if he lets other characters do the arguing that's going on in his head -- for example, Captain James T. Kirk.
The classic example of using supporting characters to dramatize what's going on in the main character's head is the Kirk-Spock-Bones dynamic: Bones is the voice of emotion, Spock the voice of reason, and Kirk the decision-maker who strikes a balance between them. To transpose this example to the Batfamily, you could easily have Alfred as the voice of reason and Dick as the voice of emotion. Or leave that behind and just have them point up different, in-character interpretations -- and remember, just because they're not arguing doesn't mean that they'll necessarily agree.
(This is one area that DC has problems with; for example, Alfred shouldn't agree with Leslie nearly as often as the writers have had him do at times, and in the latest BATMAN Jim Gordon shouldn't be so sympathetic to Poison Ivy's anti-corporatism. He's a cop, remember, and if the GORDON OF GOTHAM mini is still anything near canon, in his younger days he really hated hippies. Anti-corporation activism should seem, at best, vaguely silly to Jim. He can certainly appreciate parks, though, and that's the angle they should have used -- at least when Poison Ivy's doing something insane, the greenery looks nice.)
Further ramblings forthcoming, with examples.