Most sentient beings appreciate nuances, the lush spectrum of mental and physical aspects of existence. By contrast, an Addict places a premium on the achievement of a specific pleasure, and over a period of time that simplistic dichotomy of feelGood/feelBad displaces nuanced reality and eventually replaces it entirely.
Addiction may involve substances (food, alcohol, legal or illegal drugs), behaviors (gambling, pessimism, spousal abuse, animal cruelty), or combinations thereof (fetishes et al.).
Within a game, mechanics of an addition may be arbitrarily categorized (slight, moderate, extreme) or sliding-scale rated (no arbitrary dividing lines), the former being simplistic but more easily adjudicated, the latter more complex but more realistic.
Obviously the role played is dependent on the type of Addiction and the game mechanics used to represent the condition.
An individual with a full/major Addiction will be focused on its fulfillment. That is the most important aspect of existence, all else being secondary. All revenues will be expended in support of the Addiction, and even food is considered secondary, to a point. The character ignores or is barely aware of the negative effects, and may occasionally regret the result, but generally discounts them as irrelevant in comparison to the joys obtained. NOTE that this level of Addiction overrides the activities common to player characters in the game, and is equivalent to character retirement, being a 'wasted life' and/or termination (in the case of physically destructive addictions).
An individual with a new/minor Addiction has tasted its first joys, has taken the first step on that devastating road, but suffers few or none of the adverse/unplayable points above. All that lurks in the future, and will emerge little by little, the speed of degeneration being the primary variable. The character probably recognizes the potential negative side effects but will increasingly discount them as the Addiction progresses.
Pragmatically the game master and player should assess the deleterious effects as metrics (such as specific losses of physical qualities, if applicable) and decide in advance how far to take this before retiring the character (assuming that the Addiction is not surmounted). One alternative is semi-retirement, bringing the character back into play for a specific purpose (usually of short duration), the character being effective only as long as the Addiction is fulfilled, enabling some minor activity not focused on the Addiction (minor to that character but possibly crucial to others with a viewpoint less skewed).
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Finally, as to overcoming the Addition, it of course depends on the type and degree. You can probably find an existing game feature -- spells, counter-drugs (possibly life-long), therapy, or whatever -- to at least stop the deterioration (adjudicating unremovable permanent aftereffects, if any) and possibly to reverse those effects and even restore the whole character. In some cases a full cure may be possible. I would urge that a complete solution be found and made available, for optimal end results, unless your game is based on tragedy and/or personal angst (such as some Goth styles).
A notable roleplaying effect (in later stages of remission and after the fact) would be the character's attitudes with regard to the Addition, the substance/behavior, and toward others who are on that path. Would the character become a crusader against, or a savior of, such people or things? ...or indifferent, or some combination?