The circumstances of meeting the boss: Anarky is a boss that is connected with the side mission that concerns sabotaging his plans to bomb three buildings in Arkham. You stumble upon this character towards the end of this optional mission, after you reach the Abandoned Courthouse in the Park Row district.
The fight with Anarky and his henchmen takes place in the courtroom and it consists of three different phases. Anarky will not take part in the first phases at all so, you should focus mainly on eliminating his weak goons. Since these opponents are not too much of a threat, try and do not lose too many health points, because you will have to do with the health bar until the end of this mission.
After you make it over to the second phase, you should change the strategy a bit. Now you should stick with moving around the battlefield and avoiding stopping in one spot for longer. What speaks for this behavior is that Anarky will be trying to toss Batman explosive charges on a regular basis, whose explosion may weaken the protagonist. Also, do not forget about the anarchists with knives and do not let them launch successful attacks..
Anarky will jump down to the battlefield at the moment at which the third phase starts. Note that he comes equipped with a stun gun so, it is crucial that you do not let him hit you because you will otherwise lose a lot healthpoints. Unfortunately, you cannot react to the stun gun attacks with counter-attacks so, the only thing that you can do is leaping to the sides (at the moments at which the red symbols appear above the boss's head - the example in the screenshot) or by jumping over Anarky.
The good news, however, is the fact that Anarky is very susceptible to Batman's attacks, which he is incapable of blocking. Apart from inflicting damage to the boss, remember about his henchmen and try to eliminate them quickly. You do not need to drain Anarky's entire health bar, because the fight should end right after you have dealt with his goons.