Your main stimulus to breathe (at sea-level) is an increase in your blood's carbon dioxide content. You breathe more during faster-paced workouts and races because more carbon dioxide is being produced in your muscles and needs to be expelled through the lungs. Oxygen is all around you and has no problem diffusing from the air into your lungs.
Once inside the lungs, oxygen diffuses into your blood. This elegant process is more than adequate—at sea level, your blood is nearly 100 percent saturated with oxygen, both at rest and even while running a race. The hemoglobin of some elite runners, whose hearts pump large quantities of blood through the lungs each minute, become desaturated with oxygen when running at race pace, a condition called "exercise-induced hypoxemia." The situation is slightly different at altitude, where you breathe more to compensate for your blood being less saturated with oxygen.
Coaches often tell their athletes to breathe deeply to take in more oxygen. But since your blood is already saturated with oxygen, it's fruitless to take deeper breaths. Furthermore, since your diaphragm and other breathing muscles also must use oxygen while you run, the extra muscle contractions needed to take deeper breaths may steal some of the oxygen needed by your leg muscles. The metabolic cost of ventilation—how much oxygen your breathing muscles use—is the only factor related to your lungs that limits oxygen transport to your muscles and thus how fast you can run.