NOTE: IN PROGRESS
A guide to Catwoman by InBrightestDay ~ I haven't had much experience with her, but this is my regular setup on GH. I focus on low cooldown to compensate for lack of health, so escapes can be made faster, and also the most penetration possible for her so tanks should be no problem throughout the game.

Catwoman is a melee attack blaster, and due to her uniqueness it may take a few games to get the hang of her, but remember you are one of the squishier champions, so try and keep your distance from targets and use your Q and W to fight from range, but also use your E to escape and boost your Q/W abilities.

STOLEN POWERS

[[Flash's Super Speed]] - This is used for those escapes, it can also help in chases against faster champions with escape like Flash, but with your cooldown reduction you should be able to catch up with champions enough for a Q and W combo.

[[Poison Ivy's Healing Wave]] - This can be used in place of Doomsday's Invunerability, but it also adds additional attack damage, which can be useful endgame, although I stick to invunerability for a shield.

[[Doomsday's Invulnerability]] - I cannot stress how many times this has saved me in order to get to a healing pad, or to dish out my remaining damage.

[[Catwoman's Teleport]] or [[Cyborg's Teleport]] - Both could potentially be useful to defend, but I don't use this, it's just a recommendation.

Skills and Attacks.

Whip Strike (Q):
Damage: 55/75/95/115 - 85% of your total attack damage is applied to this.
This is the main skill you'll be using, as it has a relatively low cooldown with my set (Under 2 seconds) and has the range needed to prevent yourself being killed. If an enemy is hit at the furtherest point from the area, you will get 10/20/30/40% bonus damage. This skill can be used to kite and in lategame can be devastating againstany target, as your penetration allows you to rip tanks to shreds from long-distance.

Cat's Paw (W):
Damage: 60/110/160/210 - 65% of your total attack damage is applied to this. This skill causes the target to lose 20% move speed for two seconds, while being useful to slow down champions, it is better used in a combo with Q and E. It also allows you to close in on a target should they use a dash skill, such as Flash Attack.

Acrobat (E):
This does not provide any damage to any target, but it can be used as either an escape tool, or to add a bonus to Q or W, whichever is used first. If used before Q, it will provide 10/20/30/40% bonus attack damage, but with W, it will add a 0.5 second knockdown. Both can be used in utility, this can be used to either close the gap on an enemy, then releasing W or to start your combos off, while also being able to be used to further the gap if the enemy is too close for your liking. This skill makes Catwoman one of the best escape-classes in the game, as while it's range is shorter, it has an insanely short cooldown at endgame, which allows you to fire off insane burst-damage later on, despite W taking more time to go.

Bad Kitty (R), ultimate ability:
For 8 seconds, this adds a bleed effect to all forms of damage Catwoman gives out, which deals 60/95/130/165 attack damage over 3 seconds (gains 50% of your total attack damage). In addition, using this after Acrobat will cause Acrobat's cooldown to be reset, and the skill turns into Savage Acrobat, which deals 45/80/115/150 damage (gains 60% of your attack damage) to enemies you move through using it.

This ultimate is fucking hell if you use it after your Whip Strike, which should be boosted by Acrobat. This is when you going in close could be extremely deadly for them, as your acrobat damage could extend to over 500, not even including the rest of your combo prior and after this.

PASSIVE: Cat Burglar

Trigger: The next time Catwoman damages an enemy, a coin worth 5 credits is created, which can be collected by anyone on your team. (Cooldown 12/10/8/6 seconds) (Credit 5/10/15/20)
Passive: When a coin is collected near Catwoman, all her skills except Cat Burglar will have their cooldowns reduced by 0.75 seconds.

I've never seen much use for this skill, seeing all of her skills have stupidly low cooldown anyway, but although her passive stays the same, it could be used to reduce Bad Kitty. I recommend dumping your points into this LAST, as the main utility of this is not to get coins, but to reduce cooldown.

Items
While I usually stick to this set, you should build around your enemy team. If they're an exclusive damage type, you could probably substitute an item or two for armor or lifesteal if you feel too squishy, but I usually keep as far away from my opponents as possible, but that sometimes isn't possible.

Atomic Axe (2) and Coda Blade (4) - Both of these are so god-damn useful against a team which builds armor, but otherwise you can replace either of them with the Soultaker Katana (4) or Claw Of Horus (3) or Sword Of Beowulf (3).

[[Huntress' Crossbow (3)]] and [[Rip Hunter's Time Pack (4)]] - Both of these items, even with default amps should get you near or maximum cooldown reduction of 40%, but both of these will provide a nice amount of damage. Rip Hunter's Time Pack also has Time Anchor and Will, but Time Anchor can be used for surprise movements or to get the pesky bastard who went the way you weren't expecting.

[[Joe Chill's Revolver (4)]] - I usually take this not usually for critical damage, but for the sheer amount of damage it offers in place of other items, but since the Critical Damage on Whip Strike no longer exists, I usually will take this if I have a chance to get up close and personal with the enemy, but if I do that, [[Deathstroke's Claymore]] is also a good item to use, because while it offers a cool 50 damage bonus, the attack speed can also make the rare times you're left without will alot less painful.

Other Items you can use in place of mine.

[[Lobo's Chain (4)]] - If you need to take this item, you can substitute it for one of your attack penetration items or Huntress' Crossbow, it provides some health for Catwoman to last longer, but it also makes the occassional attack more useful to charge into, as you'll slow them down, and also be able to manuveur around your target.

[[Deadshot's Visor (4)]] - If you need additional damage, but don't want Rip Hunter's Time Pack (I use it for the Cooldown Reduction and my only Will item) or need more Will, you can use Deadshot's Visor in place, but I find the tradeoff of longer time between my bursts much less useful.

How to play this character.

For beginners, levelling your Q, then W, then E is the best way to go, then upgrading your ulti whenever you can. The best order for skill fighting is usually E, Q, W (optional Ulti at this point), E, Q, E, W, E, Q, W, using the ulti to constantly move away from the person pursuing you. This has worked as an escape several times, though you may have problems against Flash, other Catwomans, Gaslight Catwoman and champions which can get insane move speeds. This is why you can use Super Speed to keep your distance while your E is on cooldown.

Catwoman's main weapon is smart play, knowing your limits and if a target will pursue is definitely your main point of strength. Picking off targets which are slow or need to let off a skill combo to do any noticable damage will be your best targets, as well as targets which are fleeing. When I'm not capping/fighting on GH, a good thing to use is constant use of speed pads, as the speed boost will make you insanely hard to catch and can allow you to run in, deal some damage, and dash back.