Wait for the action to move to a new area #1 and start fighting. During the first phase of the battle I'd suggest focusing mainly on learning Ra's al Ghul's basic attack, the dash followed with an attack #2. The universal method of dodging these types of attacks is evading often.
After a few moments normal assassins will show up #1. Start attacking them, remembering about the need to perform multiple counterattacks #2, also those connected with the blades they're using.
Keep an eye on what Ra's al Ghul is up to. When he appears on the battlefield and starts preparing to attack, stop whatever you're doing and focus on dodging it #1. Keep fighting until you get rid of all the standard assassins #2. It should move the battle to the second phase.
You will recognize this phase by the fact that Ra's al Ghul will appear in a "slightly" bigger form #1. As you've probably guessed, it will change the whole character of the battle. Start running around Ra's al Ghul, dodging his distance attacks (mostly large shurikens) #2.
The key to winning this phase of the battle is using the Electrical Charge, though you won't need to aim at the boss yourself, as you can pretty much use the left trigger + B combination. It's very important for the charge to hit directly Ra's al Ghul and not the assassin wall he's surrounded with #1. Therefore you have to choose the right moment to attack each time. After hitting the boss for the first time an additional difficulty will appear, in the form of swords coming at Batman at high speed #2. Theoretically you could avoid taking damage from the swords by standing in the right place, but it's much better to just dodge them completely by running around the arena and evading when needed.
Constantly keep moving, regularly using Electrical Charge to drain the boss' life bar #1. After completely depleting his life, be prepared to counter his attack by pressing #2.
After you successfully counter Ra's al Ghul's first attack, start rhythmically pressing Y so that Batman can respond to his further attacks #1. Successfully completing this part will initiate a cutscene, after which the third phase of the battle will begin #2.
This phase consists of two main parts which will keep on repeating one after another. In the first, a big group of enemies will jump out of the ground. If you manage to press Y at the right moment, Batman should defend from their attack #1. Otherwise you will lose some health right at the start of the battle #2.
In the second part you will have to attack the assassins that came out of the ground #1. Defeating them shouldn't be too difficult, though that doesn't mean you don't need to use counterattacks and other more advanced moves. Similarly to the beginning of the battle, keep an eye on Ra's al Ghul #2, who will appear on the battlefield from time to time. Dodging his fast attacks is essential.
Keep repeating the above actions until you get rid of all the standard assassins #1. You should therefore advance to the fourth part of the battle, during which Ra's al Ghul will change into his bigger form again #2.
This time you will have to dodge the shurikens and fast swords from the very beginning of the battle #1, as well as the boss' attacks with a big sword #2. Therefore you have to constantly keep moving and evade often.
The tactic is identical to the one used in the second phase, you have to use the Electrical Charge (left trigger + B) and shoot so that the charges fly through the holes in the "wall" #1. Unfortunately you have to take into account that the number of obstacles will be successively increasing #2. Keep using the Electrical Charge until you drain the whole health bar.
Be ready for Ra's al Ghul to appear in his normal form #1 and start rhythmically pressing Y. you will soon have to change the tactic and use Beat down by pressing X, for a change #2.
You will advance to the fifth and last phase of the battle which will take place inside the Chamber of the Demon. Ra's al Ghul has taken Talia hostage #1 and in order to save her you will have to use the newly unlocked Batarang upgrade - the Reverse Batarang #2.
Make sure you're facing Ra's al Ghul directly. Press and hold the left trigger and afterwards RB. Wait for the enemy to be successfully targeted #1 and let go of RB, thanks to which the Batarang will automatically go round Ra's al Ghul and knock him out #2. Another cutscene will play, during which Batman will take a sample of Ra's al Ghul's blood and talk with the furious Talia. Afterwards listen to the conversations with Oracle and Robin. The next step is brining the blood to Mister Freeze.