I would consider this person driven / achievement-driven: very determined to succeed; having a compulsive or urgent quality; propelled or motivated by something; used in combination results-driven.
likewise determined: having a strong feeling that you are going to do something and that you will not let anyone or anything to stop you.
consider resolute: marked by firm determination
All three connote some character quality of stick-to-itiveness, willpower, constancy, steadfastness, doing what you are supposed to do. They are not negative in the manner of addicted, compulsive or obsessive, which can also be used to describe determined gamers.
Edit: compulsive: characterized by perfectionism, rigidity, and an obsessive concern with order and detail.
**perfectionist: one with propensity for being displeased with anything that is not perfect or does not meet extremely high standards.
Or obsessive: motivated by a persistent overriding idea or impulse, often associated with anxiety; continually preoccupied with a particular activity, person, or thing.