History
Tyrion was born the youngest and least beloved child of Tywin Lannister, who hated him because he was a dwarf. And because his mother died giving birth to him. And eventually just because. Between that and a sister who wanted him dead for taking away her mother, he had a tense home life.
The young Tyrion made do by burying his nose in books, and grew up to be very intelligent, if also drunken and lecherous. Still, Tywin saw some potential in him and appointed him acting Hand of the King during the War of the Five Kings, during which time he did a fine job of keeping a lid on the violently egotistical King Joffrey and defended King’s Landing from an attack by Stannis Baratheon. When Tywin returned to the city, he rewarded Tyrion by moving him to a smaller bedroom, forcing him to marry the teenage Sansa Stark, and putting him on trial for Joffrey’s murder even though it was pretty clear he was innocent. But when the whole world thinks you’ve killed a member of your family, sometimes it’s better to just go with the flow, so Tyrion killed his father after he escaped prison.
After his big fight with dad, Tyrion spent a season gallivanting across Essos before he reached Meereen and met Daenerys Targaryen. The two quickly bonded, and Daenerys made Tyrion her advisor. Last we saw the pair, they were sailing to Westeros side by side.
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Status
Tyrion will be instrumental in helping Dany forge the kind of contacts she’ll need if she wants to rule Westeros. As for Tyrion’s personal agenda, he’s probably feeling none too sympathetic towards his sister Cersei, who played a big part in convicting him for the murder of her son Joffrey. Jaime, who helped Tyrion escape prison following all that business, is a touchier subject. As far as Tyrion is concerned, he and his brother are in a good place, but Jaime feels rather less charitable to the guy who killed their father.
So like all the Lannisters, Tyrion has serious family problems he’ll need to address.