A highly capable administrator, Tywin served as Hand of
the King to Aerys II Targaryen for twenty years until the Mad
King, driven by envy and distrust, rejected Tywin's proposal to marry his
daughter Cersei to Prince Rhaegar and made his son Jaime a member of
the Kingsguard. As members of the Kingsguard are forbidden to marry, Aerys had
effectively robbed Tywin of his heir, which Tywin perceived as a great
insult and resigned. During Robert's Rebellion Tywin remained
neutral, answering neither the summons of the King nor the calls from the
rebels, until the Battle of the Trident, after which it became clear that
the rebels would win. Tywin marched with the full power of Casterly Rock to the
capital, feigning that he had come to defend it from the rebellion. Aerys'
advisors urged the King not to trust Tywin, even Jaime Lannister, who knew his
father would never back the losing side in a war.
Jaime Lannister was then commanded by Aerys to bring his (Jaime's)
father's head. Getting angry from this command, Jaime Lannister killed Aerys
Targaryen. Thus, since then Jaime was called as a kingslayer, or, sometimes,
oath-breaker.
To recapitulate, Tywin is characterized as a greedy feudal age
lord who can kill anyone for his power, or in other word he is always ready to
take power.