When we first meet Sam Vimes, he is a drunk, ground down by circumstances and life. The Night Watch, which he commands, and is his sole reason for living, is a travesty, ineffectual, peopled with time-servers and petty criminals, and he feels powerless to do his job, and keep order in the city he loves.
In addition, it is explained that he drinks because his natural state is "one drink below normal" -- that state of extreme sobriety that follows on from a heavy drinking session.
His family has a long history in crime-fighting, and he is the descendant of the infamous Stoneface Vimes, who executed the last King of Ankh-Morpork. The fact that the king in question was a corrupt despot has been forgotten, the fact that Stoneface was a regicide has not.
With the recruitment of Corporal Carrot and a major threat to the city as motivation, Vimes' redemption begins. He kicks the booze, marries Lady Sybil Ramkin, and begins a steady, if very reluctant, rise through the strata of Ankh-Morpork society. (In fact, it's more than reluctant, Sam is extremely anti-aristocracy, and is resentful that he is being dragged, kicking and screaming into its ranks.)
In many ways, Vimes is the antithesis of The Patrician of the city. Blunt, where Vetinari is subtle, passionate, where the other man is cold. However, the two men have a grudging respect for each other, share a deep commitment to Ankh-Morpork, and both need the other to be able to function.
Sam Vimes is the archetypal "Copper". Honest, caring (but unable to express it) and doggedly determined to get his job done.