Deathstroke
Your first fight will be inside Arkham Asylum against Deathstroke. He won't be too aggressive just yet so you can wail on him and practice your moves. There are quite a few parts of the map (such as the plant on the left and the pipes on the right) that you can use to your advantage. Deathstroke has several ranged attacks so make use of your Bat Grapple to get close or your small Bat Bots to disrupt his attacks.
If you save up your meter until the end of the fight and finish Deathstroke with it, you can get an achievement just for finishing your first fight.
Bane
This fight takes place in the Watchtower in space against Bane. There is a pretty good stage transition at the far left side of the stage as well as a switch that you can use to toast Bane with the jets at the back of the ship nearby. You can also bounce him off the console in the middle and blow up the little drones flying by above. It's best to use more ranged attacks on Bane because he's quite dangerous up close. Be careful of his charge attack. If he charges, you can use your double jump to get over him.
Minigame - Batterangs
You'll have to stop Lex Luthor with batterangs as he lumbers towards you. You'll have to hit the buttons as they appear on the lower right corner of the screen. There will be three waves but the first wave only requires one successful button press. After that, round two will require six hits and round three will require four. Lex's speed will increase for each successive wave so don't let your guard down. You will start a fight after this and the more you hit him, the more damage he'll start with. If you succeed in stopping him, he'll go down on one knee in the third wave.
Lex Luthor
This match is similarly on The Watchtower with Lex Luthor. The arena is the same so it has the transition and switch to the left, the console in the middle and the bots flying overhead. Lex is similar to Bane in that he's pretty strong close in but he has some minor ranged attacks as well. His close in moves are much slower than Bane's however so get up close and let loose on him with a flurry of melee attacks.
The Joker
What would a Batman chapter be without the Joker in Gotham. The Joker is a tricky enemy so you need to watch out for his various tells. He has a lot of variety from close to long range in a variety of paths but after he uses a move, you'll realize that he always winds up for his attack (except maybe for his magnum). You certainly want to keep up close with the Joker and really keep after him once you start hitting him. The most notable hazard is the flying car on the right side. You can blow it up and the parts can hit Joker for damage. There's also a considerable number of transitions on this map (mostly on the right side) so feel free to use those as well.
