RobH_en_tn/2ch/22/09.md

they caught him ... killed him

"they caught Ahaziah ... killed Ahaziah"

they buried him, for they said, "He is a son of Jehoshaphat, who sought Yahweh with all his heart."

They honored him enough to bury his body because he was the descendant of Jehoshaphat, who was a good king. This can be stated explicitly. Alternate translation: they buried his corpse, because they said, "He deserves to be buried, because he was a descendant of Jehoshaphat, who tried hard to please Yahweh." (See: rc://en/ta/man/translate/figs-explicit)

the house of Ahaziah had no more power to rule the kingdom

Here "house of Ahaziah" is a metonym that refers to Ahaziah's descendants. Here "power to rule" is a metonym for a king. Since all the adult descendants of Ahaziah were dead, there was no one left to rule Judah. (See: rc://en/ta/man/translate/figs-metonymy)