As far as I know, this can be done in a private match.
The Elites are Bane and Joker's minions in the multiplayer mode, which you'll play as most of the time. Like a lot of multiplayer modes now, any kind of victory earns more points, so aim to win. Here's a breakdown of ways you get points.
Kills, obviously, as well as kill assists earn points. I find the Assault Rifle is actually the most versatile weapon in the game, despite being one of the starter weapons. Short bursts can kill from a distance, full auto if you're up close. You get a grenade every life as well and you can get three more if you're first to a weapon crate marked with a grenade symbol.
Don't forget to gun for Batman and Robin, they're worth more than other minions. Avoid walking under vantage points and if there's no enemies near-by look up to see if they're in the area. If you hear a cape gliding, roll and flip around as there's a good chance they're trying to glide kick you. If you get tagged by a batarang or something similar, start rolling until you can find them, they can't use takedowns while you're rolling.
Once per match, usually around when one team has only half their reinforcements, one of the doors in the level will be useable by walking up to it and holding X for about three seconds. This door contains either the Joker or Bane, depending on who reaches it. When this happens head for the door, but don't rush it without scanning the area real quick. Heroes and other baddies will be coming for it to and this is a good chance to kill them while they're distracted.
If you get the door you can play as Joker or Bane, who both are a lot stronger than normal troops and have special equipments. Joker has a quick fire grenade launcher and a slow pistol that kills everyone in one shot. Both need reloading, so be careful. You can also kill people with melee attacks if you're close enough. If one gun runs out of ammo while you're fighting, try switching to the other gun with triangle instead of waiting on the reload.
Bane can kill people in arm's reach by just hitting triangle and holding L2 produces a rocket launcher you can use about once every ten seconds or so. He runs faster than everyone else making it easy to close the gap. His melee attack button (R3) causes him to pound the ground stunning people, but he's vulnerable while doing it, so it's not recommend unless you're trying to get someone hiding around a corner. Both villains are capable of racking up kills and all enemy troops are highlighted while you play as them.
Capturing control points is also a great way to get more points. If both teams are fighting over a central point, trying sneaking around to an unguarded control point while they're distracted. You can also tap you're detective vision on and off to get a quick glimpse of where enemies are and turn it on when you get closer to save power.
I find Bane's equipment offers some easy points over Joker's stuff. The pulse dart, which tags enemies, is great for racking up bonus points. Fire it into groups of enemies for lots of points, and tagging Heroes and Villains gets even more points. The last version of the pulse darts also stun enemies much longer, making it easier to kill them.
The UAV is also a great way to get some kills. Set a dart down first so if Batman/Robin come they'll get tagged for your allies to see. Pilot the UAV up high and aim down at enemies. This makes it a lot harder to shoot. The UAV doesn't run out of missiles, so keep firing. V2 fires faster and V4 has a bigger explosive radius. If someone is closing in on you while using the UAV, kill it by hitting up on the control pad.
Lastly, you can by buy equipment and temporary power ups from the store with Arkham Credits. I'd hold out for the most expensive container, as it's the only that has equipment that can make you stronger. The bonuses are small, but they help give you some edge, and some of the useable items can help give you a boost as well.