The study of dinosaurs is called Paleontology, therefore if you pursued this as a career, you would be called a Paleontologist.
Paleontologists actual study fossils which include dinosaurs and other types of fossils. With this qualification you could choose to focus only on dinosaurs if other fossils didn't interest you, but you may come across some very interesting finds on your digs.
In order to study dinosaurs as a career, you would need the right qualifications and training. Most employers would require a PhD level qualification. You would then most likely work for a university, college or museum, or perhaps a government funded survey and research team.