Wildlife portraiture requires patience, persistence, research, experience and sometimes a bit of luck. The research and experience will put you in the right location at the right time but can't guarantee the animals will show up. What may be a glorious location teeming with wildlife one day, may be lifeless the next. Patience and persistence allows you to capture the image when other photographers bail out. A good wildlife headshot has to have everything come together. The light has to be good, the background can't be distracting, the expression needs to say something, and the animal's face has to fill the frame. If you don't have the patience to wait for all these factors to gel, you'll go home with mediocre shots, or fewer than the photographer who waits them out.
A good headshot of a wild animal requires the photographer gets in close and has the proper gear. This translates to a long lens and a sturdy tripod. Without a strong telephoto, you need to get TOO close to the animal. The last thing you want to do is put both yourself and the animal in jeopardy. Many travelers to National Parks feel that the animals are attractions and therefore tame. They attempt to walk up to them with a point-and-shoot to fill the frame with just the face. DON'T DO THIS! If you don't have a long lens, stay in your car, as it acts like a blind. If you're lucky, the animal will approach and may get close enough to get a nice portrait. Some animals are more tolerant of people and present less danger. If you don't have a long lens, restrict your photography to these species. Start at a local duck pond where the animals have been habituated. Another alternative is to go to your local zoo.
As with the large majority of nature images, the best are made when the sun is low on the horizon, close to sunrise and sunset. Mid-day light is harsh with severe contrast issues of dark shadows under the nose and chin, in addition to eye sockets that often reveal no detail. Early and late light is soft and warm in tone, and provides a much more pleasant image. I like to use front light for my animal portraits as much as possible especially when bathed in the warm glow of first or last light. Sidelight works, but one side of the face may go too dark. If the subject is tolerant and lets you get close, use flash to fill in the shadow side of the face. Use wide apertures to help throw the background out of focus. A tight head shot against an out-of-focus background allows the subject to pop off the page.