In this chapter, Catwoman believes she can convince Poison Ivy to help her find her loot. Trouble is, Ivy's got a nasty reputation for never forgiving anyone, no matter how small their crimes...
Your first objective here is to locate Catwoman's apartment. Follow the marker towards it.
Catwoman moves differently from Batman. She's faster, but less graceful. Instead of the Grapnel Hook, she uses her whip to climb up the side of buildings, which you can speed along with timed presses of RB/R1. Catwoman is also quite weak, so I wouldn't recommend any head-to-head combat in this chapter except for the three required fights.
Catwoman's apartment is shuttered up, but thankfully for you, she can climb along gated ceilings. Get the drop on some guards hanging out outside her apartment and take one of them down with a Ceiling Takedown. The rest of them should go down easily. If you're looking for easy combat multipliers, isolating one enemy and using a Beat Down should do the trick. Once they're all knocked out, enter Catwoman's apartment to obtain her two major weapons: the Bollas and Cattraps. In all honesty, they're pretty useless in combat, but great for stealth; however, Catwoman only has stealth sections in her final two chapters, and in the Riddler's Revenge challenges, so just stick to your melee attacks and whip when dealing with Ivy.
Follow the checkpoint marker to Ivy's hideout. There are three armed guards stationed there. Stealth is certainly an option here, but if you can fight the armed guards one at a time, head-to-head combat is fine too. Once they're taken care of, swing downwards and crawl along, picking up a Riddler trophy just for Catwoman, then enter Ivy's Lair. how do you get to the joker
The fight with Ivy consists of three waves of enemies. After each wave, you'll need to Pounce up to the ceiling and climb over to the next floor, because Ivy will be poisoning the ground. Make sure to rack up tons of XP before the chapter ends.