=20pxMess with the cat, and you'll get the claws

Catwoman is a blaster with great range and variety. She can deal damage, slow, escape, chase, inflict bleed. You name it, she can do it. Her cooldowns are extremely fast and her passive makes them even quicker. Now, I used to build her much different and a lot of people grab Joe Chill's due to Catwoman being attack damage based. However, she loses a lot from the fact that skills cannot crit.

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=20pxStolen Powers
You've got 3 primary options for Catwoman in Heights. I normally grab Flash's Superspeed and Shazam's Invulnerability or Doomsday's Invulnerability. Either one works really. Atomic Wonder Woman's Speed could be used as a substitute and could help out Savage Acrobat. Your final option is to replace Superspeed with Teleport.

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Typically, my Heights set up is Speed and Invulnerability. My exception to this is characters with a dash power in which case I find Teleport more effective. Catwoman's Acrobat skill actually buffs her offensive skills, and so you may end up using it in fights rather than escapes. Go with your preference

Teleport:
Pros
-Comes with Catwoman
-Travel across the map to a turret or drone
Cons:
-Can't be used to escape or chase
-Can't help you reach the mid
EDIT: As of June 2014, Teleport's cooldown has been increased by a full minute. Catwoman's Teleport takes 5 minutes to cooldown, while the normal Teleport takes 6. That's anywhere from 1/3 to 1/5 of the match in Heights, making this an even worse option.

Superspeed
Pros
-Can be used offensively and defensively
-Can get you to the mid or points you don't control
Cons
-Best with an upgraded version
-Very short burst that doesn't get you very far (though potentially gets you to a speed pad)

Invulnerability is pretty much something you want at all times. Catwoman is squishy and it can give you the extra .1 second needed to flip out of a fight area or secure a kill in a duel.

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=20pxGeneral Strategy
Catwoman benefits most from high attack damage as an AD Blaster. While crits and attack speed affect her basic attack, her skills only benefit from building attack. I usually begin with Joker's Crowbar. Even assisting a kill will net you bonus attack, but you can get a lot of kills up top in the first round too. Even a slow start can begin netting you more attack damage as you build the Crowbar stacks up.

After this, you're going to want to look at the enemy team for where you go first:
-If they're killing you and you need to add survivability, get Lobo's Chain level 1 or 2. I'd recommend this third item at the earliest. The loss of attack damage isn't worth it.
-If the enemy team is building lots of armor already, begin the Coda Blade. You can grab Atomic axe if their tank is built full attack armor later, but I wouldn't recommend it.
-Otherwise, build Zeiss Goggles. Since the buff, their damage is insane. The crit will help when you're low on Will. And once it's maxed, Frenzy can end duels quick.
-Your third item (unless you grab a level of Lobo's first) should be Deathstroke's Claymore. Get this one to level 2 before you max out the others. Then go back to finish the rest.
-End Match: If you go REALLY late in a match, start building either Joe Chill's Revolver for the attack damage or Atomic Axe if the enemy is really high on Armor.
-Between any or all of these, you could potentially throw on Prometheum Mace. I usually don't, but it's an option. It's not a huge bonus, but given it's refundable at the 1500 level, it's better than letting your credits sit there.


P: Catwoman's passive generates a coin on a hit and reduces her cooldown when she picks up a coin. Handy, but not great in GH. It's not worth upgrading.
Q: Whip Strike: This is a nice damage dealing move that deals additional damage when you hit with the edge of it. If used after Acrobat, then it deals even more damage. This should be the skill with upgrade priority.
W: Cat's Paw is a 20% move speed slow. While it lacks the range of Whip Strike, it can make chasing easy and even helps Catwoman's allies finish a chase. If you use it after Acrobat, it actually inflicts a short knockdown as well. Note: Because of it's range, it's really not great for escapes unless you start with it. Just Acrobat/Superspeed if you need to get away.
E: Acrobat is a short flip that can be used for chases, assists, or simply gaining it's benefit. It's cooldown is about the same speed as Whip Strike so they combo very well. When Bad Kitty activates, it's cooldown is reset (and reduced for the duration of Bad Kitty) and it becomes Savage Acrobat and deals damage/bleed.
R: While her ultimate provides a lot of support for her basic, it also enhanced her skills. Cover the enemy in bleed, flip around attacks, and in general dish out a ton of damage. It's best in team fights, but can help you solo against powerful heroes as well.

Overall, make sure you're targetting squishies for the highest damage. Try to dodge skill shots with Acrobat and then unleash a Whip Strike for maximum damage (especially if you can get them into the edge of it). Your W is a great way to prevent escapes, and if you Acrobat first, you knock them down as well. It's fairy easy to win duels by staying as mobile as possible. Dodge non-auto attacks and you're golden against most heroes.

R is especially great in team fights, but use it in solo fights as needed (as soon as possible too. Get them bleeding). Flip right through them and you'll often throw off at least one skill shot before you unleash higher damage basics, whip strikes, and Cat's Paws. Even bruisers can end up really hurting if they can't hit you while you pummel them down. In addition, R resets her cooldown, which gives you the option to Acrobat, Whip Strike, Bad Kitty, and then immediately Savage Acrobat into Cat's Paw. Knocking them down, stacking them with bleed, and dealing a ton of damage.

One of the best things about Catwoman is that you don't need to fear using her skills. She has amazing cooldowns (and while some people build her with Huntress' Crossbow, I've never found she needs it). When things (especially your ult) are on cooldown, kill a drone, grab some cash, and tada, you've reduced your cooldown. Rinse and repeat until you're ready to fight. While it's possible to eat through her Will fast, your bonus attack damage, Deathstroke's Claymore, and Velocity Nines will mean you can resort to basic attacks in a pinch.