In order to unlock the "Achieving the impossible" achievement you'll need to complete the game in Impossible Mode.
Impossible Mode makes it so that both you and your enemies deal x5 times the damage: you'll be instakilling enemies, armored or not, but your enemies will be much, much more dangerous, and may be capable of oneshotting you with weapons such as shotguns. Bosses will also be more dangerous, but if you come prepared against them you should be able to kill them in seconds, before you can take significant risks. Also, picked up weapons will not come with an additional mag. Think of this mode as closer to the original gameplay of Hotline Miami.

The first thing you want to do is go to the nun and lock any weapon that is not an smg. Shotguns are the ones you really want to lock first as catching a pellet spread from a shotgun means instant death. Smgs, on the other hand, can be extremely op in your hands using Chenrezig, but still weak enough for enemies that catching a bullet or two won't kill you.

- Offensively, you should be going for Chenrezig and Homing Pigeon. Chenrezig will double your firepower, making it easier for you to hit enemies and doubling your damage output against bosses, while Homing Pigeon will turn you into a bullet hose by restocking in your mag many of the bullets you will spray. Any additional drink such as Greaser, Shredder, Prestissimo, Practiced and any you might find useful will only make you stronger. Damage upgrade drinks aren't necessary: you don't really need any more damage than you already have, so you'd pretty much only get the malus. An exception to this would be Black Market: you have enough fire rate to handle any enemy coming your way with or without it, and the double damage can come in handy with bosses. Avoid any drinks aimed at upgrading weapons that are not SMGs, obviously.

- Defensively, and this is the most important tip, you should go for Sliver (ESSENTIAL), Bubble, Halo and any drink that upgrades your Focus power. If an enemy gets the drop on you and you get hit with a burst of bullets, if a Suicide Bomber manages to explode too close to you or if you get hit with an RPG that usually means an instakill. The only way to avoid this is through Sliver: if you see your health dropped to zero IMMEDIATELY quit and reload the level. You'll get to restart it at full health. Bubble is useful as a visual cue that you are near death, and considering the damage any single hit can deal to you Halo is useful to tank damage. I recommend getting Sliver as soon as possible, especially before dangerous enemies like Suicide Bombers or Snipers start appearing. Other than that, Hair of the Dog and Bloodhound are obviously good choices for survivability, and considering the heavy usage of Focus you'll have to do drinks that upgrade your Focus such as Aperture, Rehab and Energy Transfer are near essential.

Following this guide should make your run a piece of cake as long as you use Sliver to immediately quit and reload the stage before you die. Remember also that you can quit and reload at the bar until the drinks that you need are in stock, with no need for paying the restocking fee!
Thank you for reading, and good luck!