tl; dr version
1. Take Magnum + any rounds (you only need 1 if you have 20 Gun Powder Arrows.
2. Take Bow Gun with 20 Gun Powder Arrows.
3. Take a weapon and enough ammo to kill 2 zombies (Grenade Launcher or fully loaded Shotgun will work).
4. Take the Golden Dragonfly
5. Fill the rest of the inventory with FULL HEAL ITEMS.
6. LEAVE ONE FREE INVENTORY SLOT.
7. Shoot Alexia 1 with one Magnum Round IMMEDIATELY after the cutscene where she blocks Claire's escape.
8. Shoot Alexia 2 with 20 Gun Powder Arrows as fast as you can hit the X button (took me 17 Gun Powder Arrows).
9. You should still be at Green Fine health.
10. Get the Linear Launcher and blow the agile Alexia 3 out of the air. It takes one shot to kill her, but she's difficult to hit. Don't worry about dodging her poison/acid spit attack that she does 4 times in a row.
11. Check health after every time you get hit with a series of poison/acid spit attacks. Only heal when you are at Red Danger.
12. Achievement will pop shortly after you kill Alexia 3.
General Comments for the Windbag Version
Any RE vet should know that saving the heavy damaging weapons for boss fights makes for a smoother ride through an RE game. That said, there are many bosses you can run from in REC:V because all you have to do is pick up an item and then get out of the room. Hopefully you've done that as it conserves a LOT of ammo and, more importantly, saves your heavy-hitting ammo like Gun Powder Arrows, Magnum Rounds, Grenade Launcher ammo, etc., etc.
The easiest way to beat Alexia and pop this achievements is if you have Gun Powder Arrows (explosive bolts for the Bow Gun) and that's what I'll be covering here. Hopefully you've not been too trigger happy with the Gun Powder Arrows and you have 20 left.
I have a save right before this fight and I have a TON of ammo left over for most weapons, so if you don't have 20 of the Gun Powder Arrows left, either comment or shoot me a PM with a list of what you have to work with and I will go back and count how much ammo for each weapon it takes to kill Alexia.
Items to take with you for the final battle:
1. Magnum + any Magnum rounds (you only need 1 round, but any extra will help in case you don't have enough Gun Powder Arrows to finish off Alexia's second form).
2. Bow Gun with 20 Gun Powder Arrows
3. A weapon with enough ammo to take care of two zombies (I used the Grenade Launcher as I had a surplus of ammo for it, but a the Shotgun with 2-4 rounds should be ok).
4. Golden Dragonfly
5. Fill the rest of the inventory space with full-heal items but LEAVE ON INVENTORY SPOT EMPTY as you have to get the Security File from Claire.
After dealing with the 2 zombies in this room and grabbing the Green Herb in the small area where the crawling zombie is, equip your Magnum. Once you activate the self-destruct sequence (a countdown of 5 minutes will start when this is done correctly) and leave the room that the control panel for the self-destruct is in, there will be a cutscene. Claire reunites with Chris, but this also triggers the appearance of Alexia. This wire mess floor will be the battleground where the final boss fight takes place. Claire will attempt to leave the platform she, Chris and Alexia are standing on but Alexia will block her path with fire.
Immediately after you regain control of Chris, pull the R-trigger and hit X to fire one Magnum round at Alexia. This immediately triggers the second form after just one shot.
Equip your Bow Gun with 20 Gun Powder Arrows. Once you regain control of Chris again after some cutscenes, pull the R-trigger once again and hit X as fast as possible to fire off 20 Gun Powder Arrows into Alexia 2 (I think it actually took 17 Gun Powder Arrows for me). This takes around 5 seconds and Alexia 2 shouldn't even get an attack off. If you get hit by a tentacle, just ignore it and keep rapid-firing those Gun Powder Arrows.
Alexia 3 is a flying boss now and she will spit poison/acid at you 4 consecutive times. You should still have around 3 full cure items, so we'll just ignore the poison/acid spit as this form of Alexia is agile and tough to hit.
Now it's time for the Linear Launcher. When pulling the R-trigger to fire the Linear Launcher, you will go into 1st person view through a scope.
Your health should still be fully Green Fine. Grab the Linear Launcher from its stand and start firing it at Alexia 3. It only takes ONE shot to kill Alexia's final form, but she is VERY agile and dodges many times.
Ignore the poison/acid spit, there is no point in trying to dodge this when you have limited space and you can only get off one or two Linear Launcher rounds before you have to start moving again. So just let them hit you, but keep an eye on your health after each time she hits you with poison/acid spit. This also take you out of 1st person view and allows you to re-target towards the general direction of Alexia 3. Use a full cure item when you're at Red Danger.
You'll eventually hit Alexia 3 and win the game. She can be a tough boss, but with the right equipment you can make it very, very easy. Don't get frustrated while using the Linear Launcher and missing. Just keep trying to hit her while keeping an eye on your health every time she hits you.
Again, send me a PM or just comment on what ammo you have left and I can replay the final boss and try and do an ammo count if you're out of the Gun Powder Arrows.