If you want to get this steam achievement, first you have to buy the DLC, Red Hood Story Pack. It costs £1.59. You can access this DLC at any time at the main menu, this takes 0-1 hours long.
If you want a guide here, or you can read this long text, either will do.
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( 1 of 3 ): You will start off with a cutscene. After the cutscene, you will have to take out all of Penguin's thugs and Black Mask's thugs (, reinforcements will come, ). Completing that fight will lead you to discuss with a thug about where Black Mask's gang is at. Once you get the information, you set off.
( 2 of 3 ): Once you have spawned, you will get an objective to kill all of his gang and humiliate him. Remember, the ones that look blue in detective mode can't be easily killed, though yellow ones are killed after one spray of bullets. After the brutal fight, again you will discuss with a thug and get the location of Black Mask. Let's go and pay him a visit.
( 3 of 3) After a cutscene, you will meet Black Mask, he will be firing bullets at you, so you have to dodge them and then spray a ton of bullets at him. The fastest way of completing this is to go to Black Mask and keep on punching him in the face. If a thug comes to you remember to counter or if they are charging, shoot them. After, you throw him off the window and end his life once and for all.