CHALLENGE 1 - Use Caltrops to stun an enemy (3 required)
How to complete: This challenge isn't particularly hard, it only implies using one of Catwoman's gadgets. With the Caltrops, it's worth to use Quickfire option activated by pressing the left trigger and B.
CHALLENGE 2 - Perform a Special Combo Whiplash and Special Combo Whip Trip in one combo flow
How to complete: You can approach this challenge only after unlocking the Special Combo Whiplash (A+X) Special Combo Whip Trip (A+B). afterwards locate a large group of enemies (e.g. inmates standing beside the Courthouse) and attack them, aiming for a high multiplier. After gaining a high enough value, activate one of the above skills. Quickly start attacking again so that the multiplier doesn't reset and soon afterwards activate the second skill.
CHALLENGE 3 - Complete one Claw Climb using at least 8 separate leaps without missing the timing for a single jump
How to complete: The challenge is a bit harder than it might seem at first, as the jumps need to be performed at a perfect pace. Luckily the game informs you each time if the jump was ideal, or too late or fast. After a few attempts you should master the climbing enough and you will just have to find one of the highest buildings in Arkham City (e.g. the Gotham Bank - screen above). It will let you perform at least 8 jumps before reaching the top of the building.
CHALLENGE 4 - Perform a Pounce attack (3 required)
How to complete: It's a very simple challenge, implying attacking the enemy from above. You will be informed of the possibility to do it by a claw icon appearing above his head (screen above) and then you will have to just press X. Repeat this move two more times.
CHALLENGE 5 - Use the Quickfire Whip attack during combat (3 required)
How to complete: This challenge isn't particularly hard, you pretty much just need to test one of Catwoman's gadgets. It's worth to use the Quickfire option, either by pressing the left trigger or the left trigger + Y combination (disarm).