First off you should vote "up" Punderstatement's guide because it is spot on so go ahead and do that right now.
But its good to have more than one solution to the tougher achievements like this one for people who are having extra trouble.
COMBAT CHALLENGES:
Batman has force and excellent resources in all the gadgets and disarming moves at his disposal, so switching to catwoman on the fly might be difficult for some people. I suggest doing all the catwoman combat missions separate from the Batman ones as a result.
Catwoman's combat playstyle is about using your acrobatic moves and keeping the thugs off of you. The "A+X" special move is absolutely crucial in nearly every situation.
Here are your core moves:
Strike "X"
Counter "Y"
Whiplash "A+X"
Instant Takedown "B+Y"
Beat Down "B, X, X, X, ..."
With these core moves you can 3-star any basic challenge against infantry-class enemies. The MOST IMPORTANT thing if your combo.
Variation:
This many moves used in one flowing combo will give you variation x5 for a meager 250 points (I believe). If you add-in a regular stun "B" and a Re-Direct "A, A" then it climbs to x7 for 1000 points. Add a knife takedown and an aerial attack and you have a huge amount of extra points at the end of the round.
Other moves:
Whip Trip- "B+A"--useful but I often forget to use it because its not necessary.
Aerial Attack- "B, A, A"- Use this to take away shields because when used against a shield-carrying enemy you become invulnerable during the animation...You are NOT invulnerable when you use this against an infantry-class enemy (who is not carrying a shield).
Ground Pound- "Rt+Y"- Incredibly risky but gives a lot of points--don't even try this unless you only have a few people left and have separated one enemy from the rest. This is the number 1 combo-breaker in the game so be careful.
Ultra-stun- "B, B, B"- very useful and with a huge area of effect. Use it if you remember.
Gadgets:
Catwoman has three gadgets--and there is no reason not to use at least 2 of them every round. Using one gadget gives you an extra 250 points and using 2 gives you 500 points--which is the same bonus from finishing the round without getting hurt! Using all three is a free 1000 points--and here is the best part--they don't even have to be in one combo! If you lose your combo for some reason, you might as well just use all three gadgets for a free 1000 points. You never NEED to do this but its a free 1000 points so if you get the chance you may as well...its only three moves.
-Quick Whip "Lt+Y": use your quick whip after a "Whiplash" combo move because it is reliable at a distance and you will always have time to recover and counter your next foe's attack without losing your combo.
-Quick Bolas "Lt+X": Use this second because it is even quicker--so you can use it without needing to clear the immediate area with Whiplash.
-Quick Caltrops "Lt+B": this is a gadget that might ruin your combo accidentally. It trips up enemies who come near them for a little while--and if that enemy has started to attack you, you might go to counter them just to have them tripped by the caltrops--which will register as a missed counter and your combo will be ruined. I suggest either throwing these at a far-away enemy or only if you've lost your combo already.
But the MAIN thing you want to do is keep your combo--and the way to do this is to not get ahead of yourself and use that Whiplash move to keep your space and to buy you time to think.
A good way to get into the right mindset each round is to start with a counter--wait for your enemy to approach and react appropriately. I find that if I start too aggressively I lose my combo from getting too hotheaded. My highest combo in one round is x83 so I feel my method works :)
From here you want to start striking with X, then counter when appropriate. As soon as you get the special combo move, use "A+X" to get some room and buy some time. The reason you don't want to perform an instant takedown is because A+X doesn't remove an enemy from the round--but it increases your combo multiplier. The instant takedowns should be left for later in the round when you have a much larger multiplier to take advantage of their boosted point values.
From here I typically use a gadget. Catwoman only has three and using all of them in one round (and not necessarily one combo) gets you an extra 1000 points. If you use the whip "Lt+Y" or the bolas "Lt+X" you will always have time to counter your next enemy's incoming attack--the caltrops are not a good idea.
Repeat your moves until you can use the other of your main gadgets or an instant takedown.
You want to end the match with one or 2 beat-downs. Beatdowns are worth 50 points just like an instant takedown, but Beatdowns have the special distinction of being able to raise your combo by extra. If you perform an instant takedown your combo meter will rise by x1...a beatdown could have it raised by x3 or even x8! That is an extra 100 - 350 points!
Similarly, getting a blade takedown late in the match is grants incredible point bonuses! EDIT: but watch-out, its a high-risk, high-rewards move that could destroy your combo if you misjudge the timing. (Thank you Punderstatement)
Point Values:
Strike- 10 pts
Counter- 10 pts
Beatdown finisher- 50 pts
Instant takedown- 50 pts
Ground pound takedown- 75 pts
Blade dodge takedown- 100 pts
Aerial Attack- 25 pts
Titan Takedown- 100 pts
Enemy Specifics:
-Shield-carriers: Need to be attacked with the Aerial attack. If you aerial attack a shield carrier you will be invulnerable during the animation. They cannot be countered.
-Stunrod guys: Need to be attacked from behind. Redirect "A, A" then strike once or twice, or you can stun then do the same without going into an aerial attack. You can't aerial attack a guy with a stunrod. You ALSO cannot quickwhip a stunrod carrier and this will end your combo. Be careful.
-Armored thugs: you cannot strike an armored thug or else your combo will end. You can counter their attacks without breaking your combo, but you need to stun them before yous trike them--and then you might as well use a beat-down (if another enemy gets close you may have to counter them midway through, but your combo will have risen a lot by just hitting these guys!). these are more of a blessing in disguise--they're not hard to beat and they can increase your multiplier by A LOT.
-Blade-carriers: treat these as a normal enemy, but instead of countering them, do the blade instant takedown by holding Y and back on the stick, then releasing Y during subsequent slashes. NOTE: assassins' strikes are closer together and there are four of them rather than 3. The blade instant takedown is 100 points, so get it when your combo is high.
-Lieutenants: These guys are annoying and should be taken-out first. The easiest way of doing this is by jumping over them with "A, A", strike twice, then jump over again and repeat. You are invulnerable while jumping, it keeps your combo up (even if you do it constantly), and it resets enemy attacks--so even if they had initiated a strike against you, it will never hit you if you "A, A" in time. They are also stunned by your Whiplash move so use this to your advantage.
-Titans: Catwoman cannot ride titans...learned this the hard way. Use ultra-stun, then do the same thing as the lieutenants. Remember to ultrastun every once in a while to keep them from getting you.
Any questions? Ask me!
PREDATOR CHALLENGES:
Punderstatement had a pretty good method but I posted some videos anyway because sometimes it is easier to watch.
For challenges that require caltrops, I usually had trouble getting wary guards to trip themselves. I found that if you throw them directly behind a guy he will usually hear them and turn around to check on it, but the turning counts as walking over them so he will fall down.
As far as "End of the Line (EXTREME)" goes you need a lot of luck and even then you need even more luck. It took me 17 consecutive tries to get things to work out right, and even then I had to improvise. Watch the video and take the thugs down methodically. The reason you take that first thug down right off the bat is so you can kite the enemies to that area so you can take that annoying sniper out early. Then take the enemies out in whatever way you like and when there are three or so left, use a silent takedown on a thug near a ledge so that two thugs will go to that spot. You then hang over the ledge and do the double takedown you need. REMEMBER that you can't use a silent takedown on an armored thug.