In the abridged version, he is mostly known to follow the advice of his colleagues a little too closely, such as following Nappa's (terrible) advice that he should go to Freeza and tell him to not blow up Planet Vegeta, which results of him dying at the hands of Freeza. He also shows more concern for his son as he tells Nappa to take him to a faraway planet when if things went south (which inevitably does).
Unlike the main series, Vegeta seems to have mixed feelings with his father. While he values the armor he gave him as a gift and will go into a violent rage when someone disrespects him (such as when Yajirobe calls him a "total prick"), he states that he hates him and does not like it when others (e.g. Cell) admit that he was right about him.
A running gag is when a character is trying to discuss something relating to Vegeta. King Vegeta would ask "My son, the planet, or me?", as the character who was talking to him replies with "Yes". Angered by this, he King Vegeta kills the person"
King Vegeta is stated to be the strongest saiyan of his time according to Daizenshuu 7 (excluding Turles, his son and Broly) thus making him stronger than Bardock (whom the saiyans agreed he gone mad)