Second Extinction

Here's what worked for me:

Setup
Rosy
Minigun with 3 Handling, 2 Stability, 3 Impact, and only the Scraping The Barrel perk on
Revolver with 3 Handling, 0 Crit, 3 Stability and the Reflex Sight Perk.(The Revolver Unlocks at Level 10)
Ammo Kit
Thumper
Health Stim

Mission/Strategy
Start the mission Vestige and drop at the closest drop site and follow the main mission until you've cleared all the T-rex eggs out of the cave. The T-rex should then follow you out of the cave freely, note it won't leave the cave until you've done 3/4 of the eggs.

Clear the area out and make sure you aren't being hounded by any raptors while trying to do this, squash any new nests or dinos before trying to go for the kill.

Right, shoot the T-rex up until he goes into his downed state, where he's hunched over with his tail curled up. Then let him get himself back up. Now you want to offload your minigun into the ground or whatever, so it's only got EXACTLY 149 bullets in it, 150 bullets killed it for me.

Now switch to the revolver and keep hitting it in the leg/knee one shot at a time, you don't wanna get an accidental double shot on it while it's going into its downed state. During this time more enemies tend to show up and whatnot, make sure you've cleared the area so you arnt being bothered, I had raptors knock me over and jump in front of my gun before. Also you want to be right next to his leg when he goes into his downed state.

Once you've got it into its downed state with the revolver, quickly switch to your minigun and spray its leg/knee, i didn't zoom/press LT during my attempt, I don't know if LT makes any damage difference but I wouldn't zoom in just to be safe. Once you've shot all 149 bullets into his knee, quickly call in a thumper on him.

Here's a video of me doing it earlier:


Note: If this doesn't work for you and you find it's dying sooner or later than 149 bullets, try following the same method and adding or taking away 1 bullet at a time until you get the sweet spot. That's what I did to get to my 149 number. It was 149 for the achievement, 150 to kill it with the minigun. It's literally that close.

All worries are over guys this achievement is now very easy.

Start the mission Contact
Choose Rosy as your character
Now what makes this achievement extremly easy NOW is with the recent update they added an Auto-Turret in the second equipment slot.
We are going to use that to unlock the achievement.

for your Secondary weapon choose the pistol so we can do small amounts of damage.

Now do the mission normally using the Minigun or your weapon of choice until the end where the T-Rex busts in.
Now simply kill everything but this big guy.
After everyone else is dealt with aim at the Rex and hit RB do to a callout.
This will show his health bar.
Now get him very low using your pistol and right when you think he’s about to be in his recover stage call in the auto turret and give it a couple seconds.
When you see that the turret is up and facing him go ahead and start blasting.
Get his health semi low then hold off and let your turret finish the rest. It does a good deal of damage.
it’s all about timing but if you mess up run around and wait for your turret to comeback
remember he has the brain the size of a walnut.
Juke him and then try again.

Hope this helps guys!

MAKE SURE TO DO THIS IN CO-OP WHERE THE GUY DROPPING THE POD IS NOT THE GUY SHOOTING: The easiest way to get this achievement is having a minigun with level 3 perks for damage/critical and pistol level 3 perks/mutator, use the pistol to stagger the T-rex on the mission Contact (impossible to miss the T-rex as it's a mandatory boss) and LET THE T-REX GET BACK UP ONCE, it'll be too hard to prepare yourself for his stagger HP bar on his first down. Run away and stay alive and make sure your chaingun has 149 shots in it (it should take 150 shots to flat out kill the T-rex) and then you can slowly down him the second with the pistol, when the T-rex goes down make sure you don't hit him with any pistol shots and unload all 149 shots into him and then drop your thumper into him as we have only been able to confirm this achievement unlocks with the thumper as of yet. If the rex doesn't die due to the drop pod LET HIM GET BACK UP FOR MORE ATTEMPTS. The pistol shot is just used for safety as if you accidentally shoot him once while he goes into his staggered state you can still have a good attempt at him.
Using this method A holy chainsaw, Chaosslayerx187 and I were able to get the achievement in less then an hour of attempting for it, if you have any questions feel free to ask and I'll be around to answer and will edit the solution however it may help out more people, don't downvote without a reason however.

The way I got this achievement is much like other guides already posted, however, I used the new artillery cannon gun.

Setup is as follows:
Map difficulty-
Character-Rosy
Gun- Artillery cannon (handling+2 and the extended mag) (impact +3) (stability+3 and dynamo)
Secondary gun- Revolver (handling+1) ( all the Crit perks)
Equipment- Ammo kit, Thumper, and health stim

I did this on the Contact map as the T-Rex is guaranteed to show.

First clear out all of the raptors and possible bulls. Then weaken the T-Rex until he goes into his wounded state. His tail will curl and his head will bow down in shame. LOL! Now let him recover and move around again. Make sure you have 14 bullets in the artillery cannon 15 will kill him. Once you have your gun ready with only 14 bullets, swap to your pistol and shoot the Rex until he is in the wounded state. Now call for the thumper to drop, quickly swap to the artillery cannon. Shoot! Shoot! Shot! Boom! Congrats you now have this achievement completed!