Darkest Dungeon

When a character is at zero hp(Death's Door) any incoming damage from attacks or even a bleed/blight tick will trigger a death blow check. The base death blow resist for each character is 67%, with the exception of the Flagellant class which is 73%.

Therefore the odds of getting this achievement with 5 resists in a fight for a normal character is only 13.5% so you may need to lose a lot of people if you want it to happen on its own. A Flagellant increases the odds to 20.7%, which isn't that much better. So in order to increase your odds of success without sacrificing lots of characters there are several trinkets(all can be purchased from the Nomad Wagon with the exception of the Ancestor's Tentacle Idol), a quirk, and a virtue that can increase your death blow resistance.

Tenacity Ring - Uncommon - A random drop from the Ruins +10%
Martyr's Seal - Very Rare +12%
Holy Orders - Very Rare - Crusader only +12%
Eternity's Collar - Very Rare - Flagellant only +10%
Dark Tambourine - Very Rare - Jester Only +12%
Ancestor's Tentacle Idol - Ancestral +8%

Stay away from trinkets that decrease DB resist: Gambler's Charm and Book of Holiness are both -10%. Keep an eye out for the Unyielding quirk, which adds 10% and and lock it on to your characters in the Sanitarium. Also try to get rid of the Weak Grip on Life quirk(in the Sanitarium) which subtracts 10%.

The Stalwart virtue adds +8% death blow resist in the event you do not have a Flagellant as they can never becomes virtuous, since their only 100 stress ability is Rapturous.

Therefore your best bet for this achievement is clearly the Flagellant not only because of the higher odds of success but because he becomes significantly stronger while at Death's Door(+5 ACC, +20% DMG, +2 SPD, +20% Stun, Blight, Bleed, Debuff resist) whereas other characters receive a debuff(-10 ACC, -25% DMG, -5 SPD, +33% Stress). If you heal them off of death's door they receive a mortality debuff for the remainder of the quest whereas the Flagellant goes right back to normal.

A Flagellant with Martyr's Seal(+12% DB resist, but also provides +75% DMG and +9% CRT at 0hp) is at 85% resist, increasing odds of success to 44.4%. The 'Suffer' combat skill can also temporarily increase the Flagellant's death blow resist 6-10%, but unfortunately death blow resist is capped at a maximum of 87% so the best you can do is 49.8% odds of success. Happy hunting!

This achievement seems hard to create the right conditions to boost in any way. As far as I can tell it’s random and luck just has to be on your side. So while this solution lacks a “solution”, I will add a video of what it looks like happening. You’ll need to have a character be at “deaths door” and then receive 5 more blows, but not die. Achievement will pop once this has happened. Good luck on this one.

A good fight to attempt this achievement on is the first Swine Prince boss. Since the Swine Prince does massive damage to the front line fighters, getting to death's door should be fast.

While that is being done, make sure of course to whittle down the Prince so that he dies right after one of the fighters gets at death's door and heal your fighters if necessary.

Once the Swine Prince is dead, only Wilbur remains, and he will only use his party-wide Squeal attack which does low damage and has a chance to stun everyone.

Swap fighters around or skip turns and let Wilbur do his attack repeatedly. With good enough luck, you should be able to survive 5 squeals and get the achievement.

Whether or not you are successful in getting the achievement, afterwards you may decide to sacrifice the fighter at death's door to get the ''That'll do, pig.'' achievement which requires a fighter to die from Wilbur.

2 for 1 huzzah!