The circumstances of meeting the boss: Deathstroke is the second boss character that appears while completing the game's storyline. You will fight this enemy towards the end of your exploration of Penguin's ship Final Offer, after you return from the Boiler Deck during the mission entitled Gain access to Penguin's office - Theatre.
Deathstroke is an enemy that is way more demanding than the previous one from the storyline, i.e. Killer Croc. The key to success here is a good knowledge of the use of counter-attacks, because otherwise you will fail even in the first phase of the fight, not to mention the following ones. Try to avoid taking damage for as long as possible because it is in the final phase of the fight that it will be very easy to lose health points quickly. Your failure will mean the necessity to repeat the entire fight, and not the its last phase.
The fight with this boss consists of three parts. After the first one starts, you should focus, first of all, on waiting for Deathstroke to attack, thanks to which Batman will be capable to pull off counter-attacks and land his own blows. I recommend that you take the initiative especially further into the battle, once you want to maintain a high multiplier in order to achieve more experience.
Apart from responding to Deathstroke's regular attacks, you also need to take part in scenes of wrestling with the boss, on a regular basis. Also here, you should make sure that you press the counter-attack button, BUT exactly at the moment at which there will appear a prompt urging you to (the above screenshot). You cannot press it too early nor too late because this will mean losing a given event and taking unnecessary damage. Each time after you have blocked the enemy's attack, you should land your own combination, using the fact that Deathstroke does not start defending straight away.
Perform the action described above until you deprive Deathstroke of sufficient amount of health. After the boss temporarily blinds Batman, prepare for the process of rhythmical tapping on the counter-attack key/button.
This is not the end yet, because the game will soon allow you to launch a high-intensity attack called Beatdown, which is similar to the one that you have already been using to eliminate armored enemies. Try to land as many blows, in a series, as possible. Batman should take Deathstroke's mask off and get promoted to the second phase..
The basic handicap in this part of the fight is that each time you wrestle with the enemy, you will be required to press the counter-attack key/button at two different moments (with several seconds in between) . As a result, take your time and do not hurry too much to seize the initiative. Do that only after you dodge Deathstroke's second blow.
Another new element to this phase of the fight is the possibility of a situation in which Deathstroke uses his remote claw to attach Batman to an explosive tank. In such a situation, if you want to avoid taking too much damage, you will have press the ideally timed counter-attack key/button to send the charge towards the boss (the screenshot).
Regardless of whether you managed to avoid colliding with the tank or not, it would be a good idea to follow the game's suggestion and initiate the key/button combination displayed on the screen. This will result in selecting the regular claw from the inventory. Using the claw will allow you to pull the boss your way, which will render him incapable of using his gun.
The ending of the second phase is the same as the one of the first phase so, you first need to defend against Deathstroke's blows in close combat (the counter-attack key/button) and seize the initiative and land several quick blows. Batman will deprive the boss off his blade and the third phase of the fight will start.
In the last phase of the fight, Deathstroke, as a result of having lost his primary weapon, will start using katana, which means quite a lot of problems for you. This is so, because he will be capable of initiating faster standard attacks. Therefore, you need to watch out even more and launch precise, well-timed counter-attacks. This is not the only problem, because, from now on, the wrestling scenes consist of three different moments at which you need to defend yourself against taking damage. Do not forget about this and do not attempt to respond to Deathstroke's attacks too early.
The premises of the fight's ending should be already familiar and learned in the previous phases of this fight. You first need to defend against close combat blows and initiate a very long combination of blows. As a reward for winning the fight with the boss, apart from experience points, you receive the Remote Claw.