When you get to this point in the game, you will have to beat two "boss" fights, one of them with Joker. SPOILERS included so read at your own risk.
After a short cutscene, you will be in a small room with two Titans, and several waves of other henchmen. You objective is to take out the two Titans as quickly as possible. You can do this by either ignoring the henchmen and just going for the Titans, or you can use the Titans to run into each other and/or over the other henchman to give yourself some breathing room. Standard tactics apply with the Titans - use the Batarang to slow them down, and when they stagger around, you can punch them 3-4 times before you need to back away. You can also jump on their back by pressing X from behind when they're bent over (starry-eyed) to attack the other Titan or henchmen. If it takes you too long to take out the Titans, more henchmen will drop down from the balcony surrounding the room. The only "special" henchmen you will have to fight here are the ones with Stun Batons.
Once you've beat this section, you will be taken to the cutscene with Joker. Basically, what happens here is that he takes the Titan to make himself Titan Joker, and you need to beat him (he offers to have you let the Titan take you over so two "freaks" can battle it out - you got injected with Titan by taking a dart meant for Commissioner Gordon - but Batman refuses, injecting himself with the antidote he intended to use on Poison Ivy earlier in the game).
This fight has three waves. The first two waves will consists or henchmen of all types (Stun Batons, Knives, etc) and plain old henchmen, but NO Titans. Occasionally, Joker Teeth will come into the arena, and you will need to dodge these when they play the laughing sound because they're about to blow up (they're bombs). At the end of each round of henchmen, you will be required to use the Batclaw to pull Joker down from the balcony. Each time he falls down, his claws get stuck in the floor momentarily, so all you need to do is go beat the crap out of him for a few seconds and he'll chase you around the floor for a minute (just run away) then jump back up to the balcony for the next round. The third time you pull him down, you don't actually have to beat him up to finish him.
You can relax at this point and just watch the ending cutscene and credits. Pay attention to what happens after the credits.
The Big Bang, Bigger Bang, and Biggest Bang achievements are all stackable, with the higher difficulties giving you the achievements for the lower difficulties at the same time.