Ziggurat

Using the Xbox One's Cloud Save function makes this much much easier. I was able to complete it in 38 minutes, at Level 11, with the Magma Rifle and Scarab Staff by using this. Here's a quick rundown of how best to go about it.

Start over and over until you get a weapon you like at the start (I recommend a strong Staff or Cannon - the Magma Rifle worked fine on the final boss, but I also had the Amulet that increases Alchemy RoF and Level 3 of the "Increased Alchemy Damage/Other Damage types Decreased"), then just complete the entire first floor (as long as you're under 7 minutes you're fine). When you load into the second floor, check the Weapon immediately and keep whatever Weapons you think are best (the one at the beginning of each floor will disappear by doing this, so always check them when you load in and keep the best).

Now for the annoying part - Save and Exit back to the main menu, then completely quit the game (make sure it's closed down). Then go to your Network Settings and choose "Go Offline" (it's on the left side of your screen, and is the 3rd option) - this will sign you out of Live, but leave you in your Profile. Now start the game and load your save. What you want to do is completely explore the floor - look for the quickest way to the Key and Boss room then either remember or make a little map on paper (I'd mark any Chest or Danger Rooms that are easy to get through, so if they aren't more than one room off track then I'd consider doing them).

Once you have a route established, just close the game entirely (no need to Save). Then press the Menu Button on your controller over the game and go to "Manage Game." Go to the Save Data section and click to delete the Data (it should say "Delete from Console" when you click on it), and if it doesn't delete then do it again (sometimes takes 2 goes). Delete everything in the "Save Data" tab, doesn't matter what's there. Once it's clear, go back to your Network Settings and choose "Go Online." Give it a few seconds to reconnect (up to 15 depending on your connection), then start up the game. It should say "Syncing Data" on the main menu, which means it worked. After it's done syncing, close the game down completely again, then go offline again.

The layout of the floor your on will be the same, but the monsters in the rooms and any items from Chests or Level Ups will be different (so be careful, the monsters can be radically different). Follow your ideal route, and finish the level - but this time will be different slightly. If you are satisfied with your run (anything under 8 minutes is fine, and you should be able to gauge how well you did), then pause your game BEFORE going to the next floor. While paused, go to your Network Settings and choose to go online again (don't worry, your game timer is paused too). Let it connect again, then go back to your game. When you're connected, go through the portal - then repeat everything all over again (check the gun first, then Save and Exit, close the game, go offline, load your game, map your route, close the game, delete all in save data, go online, sync your save, close the game, go offline, and ideal run).

The game will tell you how long it took you on each floor, but you have to be online before you get to see it by going through the portal - so try to gauge it yourself (and keep track of the times it gives you after you enter the portal so you know what you have left). My times were (roughly) 5:30/7:30/7/9/9.

If you die at any time while offline - quit your game and close it completely, delete the Save Data completely, go online, start the game to resync your data, close the game completely, go offline, then try again. You have to close the game down between syncs, and you have to be online when you hit "Save and Exit" and then close the game down for your save to be updated on the Cloud. If you just go on and offline without closing the game, then your data won't update on the Cloud.

Last thing - when you beat the final boss, you'll be offline. Get to the Score Screen to see your complete time. If it is below 45 minutes, then go to your Network Settings and choose to go online (while leaving your game on that Score Screen). Once you reconnect, go back to the game and get back to the main menu - the achievement should pop right away (it won't pop while you're offline, so don't panic).

I know this was a long post, but feel free to ask any questions. This same strategy works for a number of things. I'm using it for my Hard completion as well, and I used it for an achievement in Little Nightmares where you weren't allowed to die. Just make sure you follow the process exactly - go slow if needed (a single deleted save at the wrong time or accidentally dying while you're online can ruin your run).

I kept dying on the boss until I got the perk Endure, which makes it so you can't be hurt for a short while after taking damage. This, in combination with several ranks of Thick Armor, made it so I could tank most of the damage.

I would ignore any perks that won't really help in the long run. I never really went for Druid (Health potions heal more) because this doesn't actually help in the boss fight, and depends on drops. I would sometimes take Wrecking Ball just because before a boss fight you can run around breaking stuff to get health. Generally ignore any instant perks (any that just does an affect as soon as you take it). These are generally perks that just instantly heal you or something. I'd avoid them unless you're desperate.

The weapons I used were Frozen Soul, Eagle Staff, and Frostbow. Most staves are pretty good so you don't necessarily need that one, but the other 2 abilities are fairly powerful. I also had a Golden Amulet, which makes you attack more quickly with your Alchemy skill (Frostbow) for a short while.

The 45 minute time limit itself should not be much of an issue even when exploring entire floors. My worst time where I actually made it to the boss (and died) was 50 minutes. If you have these weapons most rooms should be a quick sweep.

The main things you want to shoot for are the Endure and Thick Armor perks. Your weapons shouldn't matter too much as long as it's anything with good accuracy.
Frozen Soul's 2 firing modes are essentially the same, but the alt-fire is a tighter burst that does less damage with a guaranteed freeze (though it cannot freeze bosses).
The Eagle Staff homes in on enemies, with the alt-fire firing more shots. I also like the Eye of Twilight. Essentially any staff that is accurate.
The Frostbow is just a very powerful weapon, on par with if not better than the Magma Rifle. Any single-fire gun/bow should work.

Edit - Since this post I have yet to ever get the endure perk again, so I don't know how rare it is or if I'm unlucky. There is also an amulet that makes you invincible for a short time that could fill in for it.

From Lv 1 Blue Slime "Mirror of Fate. It puts the mystery box, portal key, and boss room on your map for every floor. I haven't gotten around to a speed run yet but I'd guess that perk would be very helpful for it."
Personally I cleared out all but the last floor, so time isn't too much of an issue.

I tried for years to get this achievement. I finally realized the trick is to change the controls so the d-pad moves your character instead of the left stick. This causes quicker speed by default, as this game seems to be designed with traditional asdf keyboard controls in mind.

After I changed the controls I got the achievement pretty quickly.