My father asked me if I could get along with the Jones family.

Literal

Father [topic-は] me [to-に] Jones's family [with-と] well doing [can-ことができるかどうか] asked.

うまくやっていくことができるかどうか — can (one) get along well? Embeds both ことができる (can do) and かどうか (whether or not) in an indirect question.