Here's what worked for me:
SetupRosy
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/StrategyStart 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.