In order to get a dinosaur to 5k or more rating, the only way to do this is by gaining rating thru combat. Every time a dinosaur kills another it will gain 50% of the losers total rating.
So i will explain how i managed to get this.
I did it with the indominous rex and diplodocus. But technically this can work with any other carnivore. The reason i used the indominous is because its the only predator that i believe will attack and kill sauropods (long neck dinosaurs)
1. Go into Isla nublar Sandbox mode (you need to have this unlocked by playing the campaign) we do this there due to the unlimited funds and that we can tinker with the game settings. For example turn off escapes.
2. create several pens, i did 4, but at one point i was thinking of doing 6 or 8. The more you have the faster this will go. They need to be large enough to be able to properly house the indominous and the sauropods. With food and water.
3. In each pen Incubate an Indominous rex with aggressive instincts and any other traits you wish to add to them.
4. Start incubating Sauropods as fast as you can push them in each pen with the indominous rexes. Try to have them with high ratings too. I tested with all the sauropods and got the best results with the Diplodocus. For some reason they attacked those over the rest of the dinousaurs.
5. The Indominous rexes will kill the dinosaurs and gain around 100 infamy rating on each kill. You have to be patient and keep feeding the indominous dinosaurs. Also make sure everyone is healthy with enough food and water as starving or dehydrated dinosaurs produce a lower rating number.
6. The slow method. I assume you can just keep feeding them dinosaurs until one eventually reaches 5k. I got one indominous to 3.2k just by doing that. But if you want to finish faster. Go ahead and when your indominous rexes have 2-3k in rating, have them fight each other. Assist the one with the higher rating number to make sure it wins.
This Took me around 2 hours. But i got there using this method. I also tested other methods like using smaller herbivores, other carnivores. Or even pitting T rexes and other large carnivores against each other and they all proved slower than this.