Off the top of my head, some quick thoughts on the Alpha aspects of some characters. There's certainly some stuff to takeaway from them if you analyze them in this way.
Ned Stark - Alpha. Probably the most well rounded embodiment of the traits. Has a code, sticks to it, owns all his responsibilities.
Robert - Alpha in regards to his own pleasure, to the point of being self-destructive. Definitely lacking in a lot of areas.
Stannis - Alpha about justice and ruling, but his at-all-costs mentality is questionable.
Jamie - In regards to being the best at using a sword (before he lost his hand), he's Alpha. But he's got serious oneitis with his bitch sister.
Tywin - Alpha, but he cares too much about how others view his family.
Tyrion - He derives too much of his status from his family name and money, but he demonstrates the necessity for men to focus on improving what they have (his mind) over lamenting what they don't have (good looks, a normal body, etc) Alpha in that sense.
Ned Stark - Alpha. Probably the most well rounded embodiment of the traits. Has a code, sticks to it, owns all his responsibilities.
Robert - Alpha in regards to his own pleasure, to the point of being self-destructive. Definitely lacking in a lot of areas.
Stannis - Alpha about justice and ruling, but his at-all-costs mentality is questionable.
Jamie - In regards to being the best at using a sword (before he lost his hand), he's Alpha. But he's got serious oneitis with his bitch sister.
Tywin - Alpha, but he cares too much about how others view his family.
Tyrion - He derives too much of his status from his family name and money, but he demonstrates the necessity for men to focus on improving what they have (his mind) over lamenting what they don't have (good looks, a normal body, etc) Alpha in that sense.