For STARE INTO THE ABYSS you have to finish Scene 15 without dying on the 3rd floor, then grab the floppy disc in BOTH cutscenes (as Jake and Evan). Once that's done you should unlock the bonus level and the achievement.
COMBO GOD can easily be done on Scene 12 (Death Wish) as Mark on the first floor, alternatively can be done on Scene 17 (First Blood) by rewinding until the first enemy has an automatic weapon.
FANATIC can be done on Scene 1 (Down Under).
FAMILY BUSINESS can be done on Scene 21 (Seizure).
SNAKES ON A HEAD can be done on Scene 3 (Hard News).
If needed, you can break electronic devices on Scene 3 (Hard News) for PERCUSSIVE MAINTENANCE.
A+ Ranks are more lenient in this game compared to Hotline Miami 1, so you don't need to full combo every floor. That said, combos are still the way to go, so you should try to combo whenever you can.
If you plan carefully, you can leave the last enemy on a screen near the exit, allowing you to carry the combo over to the next screen. Doing this pretty much guarantees an A+ or S Rank if you keep the combo going just a bit more.
You can bait enemies to you by either firing a gun closeby, or by peeking around a corner so they see you and run to where they last saw you, or do the same but behind glass. This is useful for keeping a combo going, and a safe way to take out enemies.
For the Fans levels, Mark is the best choice due to the high amount of starting ammo.
For the Beard levels, Flamethrower is the best choice because it's low-risk high-reward, and it has the best ammo recovery (40 ammo on restock).
For the Jake levels, Irvin mask is the go to since you get a nail gun with decent ammo, and it's silent.
For the Son levels, Bloodline is the best choice due to the high amount of starting ammo. You will have to use up the rest when you have to face the sword guys unfortunately.
For the Writer levels, don't forget that de-arming guns not only gives score, but also counts towards your combo. Always carry a blunt melee weapon with you.
Hard Mode flips the level vertically, adds more enemies (including fat/sword dudes), and halves the ammo of guns you pick up EACH TIME you pick them up. Therefore you should not recklessly cycle through guns, but use them up before moving on to the next one. Take it slow and bait enemies around corners a lot, it can take a good amount of trial and error. Always remember to bring a gun with a good amount of ammo to the next screen if possible.
Select Scene 17 (First Blood).
Rewind until the guy next to the door has an automatic weapon.
Run in and shoot as many people as you can except one guy close by so he can kill you.
Ideal set up is that upon pressing R, you have a 5 kill combo and die instantly. Getting the proper timing can take quite a few tries. Rewind if you didn't get the set up.
The more kills you have upon dying, but especially the time between each death (ideally, instant) will greatly speed up the process.
Once you have the set up in place, run the script below until you're done.