There are very few people out there with that job; it's an extremely tough discipline to break into. That said, here's what I'd work on to start down that path:

1) you need to be a good photographer. More than that, a good nature photographer, with a portfolio that stands out. So, go out and shoot. Shoot at zoos, shoot in refuges, shoot wherever you can get out work with animals in their environments (heck, shoot ducks at the local park to learn how to shoot things that are moving and not cooperating and may not want you to be near them). Shoot a lot, build an online portfolio of your best.

2) Study nature and the animals in it. You'll need a background to understand the species you're shooting and the world they live in. Very often the big difference between photographers with average shots in the field and the great ones is that the great photographers have done all the study and know what to expect and how to plan their way to improving the odds for that shot. The better background you have in this, the better you'll understand the subjects.

3) Volunteer. Find local nature organizations you like and start volunteering. The local humane society. The local wildlife rescue groups. Look around for organizations you can be helpful to. You amy well start out cleaning cages or feeding recovering animals, not being their photographer -- but get involved in the organization, and when you go, carry your camera. You won't get the photography jobs to start because lots of people want to be the group photographer -- but find ways to contribute and get involved, and you'll find opportunities to take photos along the way. Give those photos back to the group, and over time, they'll start asking you to do that. This kind of work can open doors that won't open any other way.

4) Make sure you're comfortable roughing it outdoors. Camp. Hike. Learn to backpack and hike out there. One thing people tend to forget is that if you're going out after rare and wild species, there are rarely hotels nearby, so you need to know how to handle yourself in rough terrain in bad conditions, and how to hike yourself in and out of the locations where those species are. And don't forget, the photographer who took that shot you drool over in National Geographic was probably out in the field tracking it for a month, not a weekend. It's not an easy life, just a rewarding one.

Those to me are the cornerstones of a foundation of a career that can lead in the direction you want to go. Not easy, not overnight, and maybe not ever. But if that's what you want to do, it'll be the kind of things to do to get you going down that path.