You can only purchase this mutation immediately after exiting Prisoners' Quarters, i.e. if you pick something else then proceed to a level 2 biome, you won't be able to get it on that run.
The quickest/easiest route I'd say is:
Prisoners' Quarters [30 kills / 2 mins] > Promenade of the Condemned [60 kills / 8 mins] > Ramparts [60 kills] > Black Bridge (The Concierge) [Flawless / 15 mins] > Stilt Village [60 kills] > Clock Tower [60 kills] > Clock Room (The Time Keeper) [Flawless] > High Peak Castle [60 kills] > Throne Room
With the requirements for the Timed/Perfect doors noted above, if you're going for this achievement you may as well just play on 0BC anyway as well.
If you're ever playing cautiously as well (particularly to get X kills without taking damage for the Perfect doors) I'd recommend a Tactics build. The "Double Crossb-o-matic" skills put out a lot of damage and take aggro from enemies. To help get the you want you can (before the run) locate the Architects Key hidden somewhere in a wall underneath the Graveyard and use that to unlock the door on the far right of the overground portion of the Graveyard. In here is the Merchandise Categories unlock (costs 300 cells and requires you to have previously unlocked 30 other items) that will let you select what type of items will be available for sale at shops, to get more relevant gear for your build.
Going into Tactics will leave you a bit low on health, but if you come across any Minotaur's scrolls (that only let you upgrade Brutality or Survival) this should boost it up enough, as you shouldn't be taking many hits anyway.
Taking it a bit slow chucking out the crossbow turrets should make it a reasonably easy run, if you get Hunter's Instinct (decrease skills cooldowns) and Support (increase DPS when nearby deployable skills) for your other two mutations you should be able to have pretty constant uptime on them and a good boost to their damage output.
Weapons are really down to preference, I wouldn't say there's anything outside of the ranged weapons that boosts with Tactics that's worth considering. I'd always try to go for something like a Multiple-nocks bow and a Frost Blast they're good for keeping you out of reach/preventing enemy attacks so your turrets can finish them off. Try and get status ailments on your weapons that harmonise with your turrets too (e.g. weapon causes burning, turret has +100% damage to burning enemies). On your Amulet, and shield if you want to trade one of your weapons out for one, try and look for/roll for a good -x% damage taken affix.
Don't forget as well if you're being particularly cautious that you can always use the Homunculus rune to safely grab items/kill enemies (even if it is really slow), and your dive attack is useful for clearing out single-hit enemies, and just slowly whittling away some health/jumping back out to safety.