This is pretty simple, and can tie in with getting other achievements [Hey, Big Spender! and Keep Your Balance].
First off, however- information that I should have originally added in here and didn't.
Femme Fatale is NOT available right off the bat- you need to complete the game on "Fixer" first to unlock it. [Thanks L0cke89 for reminding me of this in the comments.] Also, if you do play the game on Fixer first, you WILL unlock the achievements for the lower difficulties. Now with this out of the way, on to the actual guide:
In the first level, Chinatown Rundown, try to score as many Style Points as you can without doing anything too risky. Basically, jump into the air, begin to fire- but if you want to get more points, tap the "B" button to go into a slide and continue shooting, Tap the "Left Trigger" if you slide up towards a wall to run up it and backflip off. Killing while you're in slow-mo during these segments scores you big points.
Which, is going to be helpful and help you breeze through this difficulty a little faster.
When you get to the first Arena, grab all of the multipliers you can while killing the thugs- but DO NOT close any of the spawn doors. Remember to keep jumping/sliding [Seamlessly, tap "B" while you're STILL in slo-mo in the air so it will transition without the loss of your combo] and wall-running, if possible, to score extra points.
If you grab all of the combo multipliers first, you should quickly be able to earn a 5X Max multiplier, which will also help you in your quest with the "Maximum Efficiency" achievement.
As long as you DO NOT continue to do the same move over and over, let's say, sliding, for example- you will at least earn 50+ points for every kill. If you string different moves together as you kill, you can earn upwards of 150-200 points for each person you kill.
If you do continue to slide over and over to kill, you will lose points and at times will only get 8 points or so for each kill. It's not very helpful.
When enemies start to thin out and no longer come out of the doors, close them all. You will get 75-100+ points for each, but make sure enemies are no longer coming out. I've had first-hand experience where enemies would stop coming after waves and waves of them, but as soon as I closed it the spawn door, one enemy would hop out just a second before it shut.
When the Arena mode ends- depending on the Length of time it took to finish, how many you killed while in Acrobatic slo-mo, AND the combo meter [Which should always be at 5, or near-full at 4]- you will be awarded extra points.
If you wait for every enemy to be defeated before leaving, there is no likely way you will get "Excellent" for every category, which is fine. If you've killed most of the enemies in slo-mo, and kept your combo up, you'll be rewarded at LEAST 1000-1250 points as a bonus.
Now, when you are purchasing Upgrades- FIRST, go for Pistol Firing Speed- upgrade that twice, and also upgrade the Damage once or twice as well. Depending on the amount of Style Points you earned, you may not be able to fully upgrade them.
Make sure you have enough points leftover to at LEAST buy an extra block of health or two for Rubi in this area, too.
When you continue on, and get to the next Arena- first, GET AWAY FROM THE CARS! Shoot them ALL when you are far away.
This will keep you from accidentally blowing yourself up when you're jumping over them, and may hurt some of the thugs trying to kill you.
When you've got some of the cars destroyed, begin to kill the thugs, and try to run for the Combo Multipliers floating in the air. You do NOT need to swing on the poles to get them, just jump directly under them- you should pick them up.
Continue to destroy the cars, and mix up your Acrobatic Slo-Mo movements as you kill the thugs. The best way to do this is possibly have the auto-lockon gun on a thug, and blast away the cars with the other. You can kill, and get the flaming wrecks out of your way.
Continue to milk the spawn doors the same way you did in the previous mode, and try to slide up to walls -AT AN ANGLE- to enable you to wallrun, before you run across them, and JUMP off them into an acrobatic dive/roll- you'll get a LOT more points this way.
If you want- you can try to add another slide in, however- I am not 100% sure this will give you more points, since it's being used in the same combo. It may give you less points, so beware.
When you finish this arena up, once again as long as you killed most of the men in Slo-Mo and kept your Combo up around 5, you should get a 1000+ Point bonus again. Buy more upgrades. This time, try to finish the rest of Rubi's pistols.
The reason I am so dead-set on these pistols, is because they are the only weapons you will need in this mode. They do not need to reload- you simply can just hold the fire button down for eternity and kill, kill, kill.
If you have left-over points after finishing her pistol upgrades, go for her acrobatic upgrades.
Fire from poles, fire from ziplines- the upgrades that allow you to fire while doing something. These are insanely helpful- but I will warn you about firing from a poll- TRY TO AVOID THIS WHEN YOU ARE IN AN ARENA OR NEAR A LOT OF MEN!
They will ALL have amazingly-accurate aiming when you're on a pole, and will whittle away almost all of your health in no time! Try to avoid swinging from poles around a lot of enemies, stick to basics!
When you get to the freeway chase- this is where the pistol upgrades ALSO come in handy, as does Rubi's upgraded health.
She can take a few more bullets, and that will be very helpful as you will not be able to avoid taking some shots.
But with her upgraded pistols, aim for any thugs head that pops out of a black car/van.
Now remember- you don't need to get this level done in one full run, so don't get too frustrated. When you get to the Quick-Time events, when you pass them- you have reached a checkpoint, so it's alright to die if you're not sure you can finish the level without more health.
When you reload, you'll be at the point where the Quick-Time event ended, and you will have full health. [This is also true for Chapter 11, Road Rage.]
Remember to keep aiming for heads- and fire like a crazy man [Or woman as the case may be] when the game prompts you to jump from car-to-car. You can do some incredible damage this way, or pick off annoying enemies with the lock-on gun. You will also clear enemies out faster with the upgraded weapons, which may make the level easier.
However, if you are not able to get headshots- just aim for the body. This level is really just about making it through, and the extra five points you get for headshots really isn't worth it if you're not getting anywhere.
When you get to the end of the level- if for whatever reason you haven't finished upgrading Rubi's pistols- DO IT NOW! Everything should be full on her pistols, and you should also have enough Style Points at the end of this level to purchase the rest/last [depending on how many you bought so far] of her health bricks.
Now, at this point you should have all of these upgraded/purchased:
Rubi's Pistols: Firing Speed, Bullet Damage [Should be maxed out at this point.]
Rubi's Health. [All of it should be purchased, too.]
Firing From Pole/Zipline, etc. You only need the "Fire From..." upgrades- everything else can be left alone, for now.
If you have all of these upgraded- you should be okay for the rest of the game. That sounds odd, yes- but as long as you do NOT purchase any other upgrades- you should be able to get the "Keep Your Balance" achievement before you even reach Chapter 10, and can splurge it all to unlock "Hey Big Spender!" directly afterwards.
When you get to Ratboy in level three, he is fairly easy- ignore his men, if you just want to kill him- and fire away. Fire, fire, fire, jump, slide, jump, continue to fire. Very soon afterwards he will turn and run- maybe a minute or two of straight, non-stop firing.
If you want to kill his men for extra points- go for it! As far as I can tell, they don't stop spawning for a little while. It may get boring- and dangerous if you just ignore Rat Boy- he can do serious damage with the Minigun.
I have two final pieces of advice to help you get through this game, and those are:
On Chapter 4, "Falling for Kafka"- ALWAYS FIRE! DO NOT STOP! Even when you see no enemies, CONTINUE TO MOVE THE CAMERA AROUND AND FIRE BLINDLY! This will help you SO much- I finished this level on my FIRST try on Femme Fatale doing this.
As for the free-falling part, I cannot understand why anyone has a problem with this. It's simple- look for gaps in the pieces of the airplane to manuever through, and avoid the boxes. Also watch out for the QTE at the end of the level, if you forgot for some reason.
The second tip- The enemies with the miniguns, they are annoying and easy at the same time. In Arena modes, try to have the Auto-Lock gun aim at the man with the Minigun, and ALWAYS FIRE!
Try to pick off the regular thugs with your manual-aim weapon, and until you kill the Minigun-man in these sections, it is okay to continue to jump over and over and over to get him.
Also, if you are stuck at an Arena with a Minigun-user, and it's past the point where you unlock the Crossbows, fire blindly at him with them. They will lose health VERY fast- even with no upgrades to the Crossbows. They will fall quickly, and it might save you a lot of time and frustration.
If you don't have the Crossbows unlocked yet, Shotgun or SMG's are the obviously only, and acceptable alternatives. Of course, you can just exclusively use the pistols- it's your choice. Whatever you are more comfortable using.
Also as well- I know The Ice House is a nightmare, the Arena.. it took me thirteen minutes to complete it using the tactics mentioned above. It is a nightmarish arena, and my advice is just try to wipe out the Minigun-user ASAP- AND AVOID COMING INTO VIEW OF ANY OF THE MEN ON THE PLATFORM ABOVE THE ENTRANCE TO THE ROOM IF YOU'RE NOT SHOOTING AT THEM!
Try to just get the 5X multiplier by killing non-stop here, that's what I did and scored many, many style points in that section.
If you have any questions, or are confused by a certain part in the game, gameplay or plot wise- send me a message, I'll be glad to try and help with something not mentioned here- or something you're confused by in the guides above.