Here's what worked for me:
First, having the difficulty on Aspirin is definitely a big help.
Next, make sure you play this through a few times so you know how many guards you're dealing with, where they are, and even watch that video above to try to count how many guards he's killing. I tried going along with the video but I kept failing.
Anyway, once you have an idea of how many guards and where about they are, you need to understand that no one is allowed to die, and that Lynch needs to STAY PUT at whatever safe spot you lead him to. You do that by looking at a spot and hitting B. You know you did it when you see a white shield pop up on that location.
Anyway, the game allows you quite a good distance before you need to get Lynch closer to you, so keep that in mind.
What worked for me in the first part after grabbing the girl, is after taking out the 1 guy on the rooftop (via the windows in the girl's office, above your head when the cutscene ends), taking out the 2 guards on the walkway just outside in the club area, on the 2nd floor with you. After you take them out, there are 2 stairways, one to the left and one to the right. Make your way ahead and to the left, as there is a guard there who may or may not be alerted by the sounds made while killing the previous guards. Whether or not he is, as long as you're on higher ground, if he tries to shoot you, he won't be shooting in the direction of the civilians. If he's alerted, make your way back to the office area to lure the alerted guard to the walkway where you previously took care of the 2 guards.
There's another guard down the previous right stairway, at a pillar. I took the left stairs and circled around to get to him, and killed him there. IMMEDIATELY after that, run back up to the office area.
The goal here is to lure guards a little bit at a time back up to the walkway where there are no civilians. Guards have a pretty decent field of vision. The key to realizing when a guard spots you is to first scout them from a distance, see where they are, track them, and watch their flashlights. As soon as their flashlights stop circling or moving around, and point directly at you, you know a guard is making his way to you.
While it sucks that you can't really run, if you take it slow when approaching guards, you will know the instant they spot you, and you can make your way back to that upper walkway. After you clear out all the guards in the main room, and the few in the bar area (there's a couple there, and one at the entrance which is around the corner) make your way to the stairs in the bar area that lead you to the checkpoint mentioned in the video. Bring Lynch a little closer, but not to the stairs yet, otherwise guards will be alerted even though they can't see him.
After the checkpoint, take care of the next few guards before bringing Lynch to the spiral stairs to your right that take you to the 2nd upper area. Clear out a guard to your left and to your right, hopefully they don't misfire and hit a civilian, and proceed to a few more guards up ahead that may be heading your way.
You may choose to lead guards ALL THE WAY BACK to the original walkway if you want to be safe, and that's perfectly ok. The achievement requires patience anyway.
After you make your way to the rooftop exit after dispatching the next few guards, the rest of the achievement is much easier, as it then just becomes a point of making sure Lynch doesn't die. You can follow the video for that part.
The video I've linked is not mine, and is the same video linked in the guide posted by MoonDoggie82, however I did not follow the video for the first half of the nightclub. I used the walkway luring.
Hopefully that helps for anyone whose having trouble with some of the other solutions.