This is actually really easy; the hardest part is collecting enough scrap to actually build the Archangels.
All but one Death Run location has 1 Archangel that you can drive for that particular race (The Reconvene uses 2 Archangels).
I would recommend having a fully upgraded Thunderpoon and simply blowing up all of the opposition as soon as a race starts; that way you're also working towards The Exiled
. Once you're out of ammo with that; you can either try a race and use the shotgun or harpoon instead or, if you have a stronghold armory upgrade, just fast travel to that stronghold and replenish your Thunderpoon ammo. (NOTE: Sometimes the ammo won't refill fully after visiting just one stronghold, so you may want to go to two)
There doesn't seem to be a penalty for failing; so if you think you won't make it to the finish in time, just hit the
button and get out of your car for an auto fail.
Once you've went through all 16 Archangels; the achievement will unlock.
Side note: Storms can still happen during these and they can make it a pain; if one happens to hit you during a race, I would strongly advise you quit the race and try again later.