The Heroes
Unfortunately, the game assigns what characters you can play with on which levels early on, and it's only a duo. In the beginning of the game, Superman and Batman have their hands full taking out the deadly robots wreaking havoc on the city. Smashing the robots and saving lives is the way to go. The best approach is to toggle back and forth between the two heroes. Batman offers his long range Batarang attacks and cape shield, which is a perfect counter-attack to projectile-firing enemies. In closer combat, go with the Man of Steel. He heals faster, and with enough power-ups becomes invulnerable.
Episode 2 offers Martian Manhunter and Zatanna; MM is pretty much the best bet through all levels, as he can temporarily phase though enemies, his mind blast takes out the baddies in one hit and he is much tougher than Zatanna and her magic spells (she's good to use for her healing spell though). As with Batman and Superman, Green Lantern and Flash work best when toggled. Flash's speed bursts are short but can clear a room, so use them wisely. Green Lantern's ring can handle about every enemy on any level, but he's somewhat injury prone when not protected by the ring shield. Wonder Woman is second in strength only to Superman, so choose her (again) over Zatanna, although her magic time shift is best utilized with a screen full of enemies. When Superman and MM are together, they're very similar so simply pick your favorite and go with it.
Once you've made it to level 6, you can choose any of your unlocked heroes; go with Green Arrow, as his ranged attacks allow him a safe striking distance. On the following level, you can choose whatever three characters you prefer, so go with the ones you've had the most success with. Keep this in mind for the last few remaining levels, as the difficulty increases quite a bit, and you'll need the familiarity with your character's powers and moves. The logical choice would be the big three: Superman for strength and invulnerability, Batman for gadgets and projectile fighting and Wonder Woman for strength. Zatanna makes a good choice for her healing powers as well.
The Villians
There are numerous minions to depose of in any number of ways, but boss fights present a tougher challenge. Resist the temptation to get too close to a boss character, as the game developers made them extraordinarily tough, even against a powerhouse like Superman. Use projectile fighters like Huntress, Batman, Green Arrow or Green Lantern to weaken the bad guy, then toggle to a strength character like Superman, Manhunter or Wonder Woman to get in close to deliver more lethal blows. A few moves like this takes out most bosses. This team strategy is especially important when confronting Darkseid, the game's final boss.