How should I ask my dad if 'I have Korean in me' in Japanese?

why the need to ask in japanese? anyway,

watashi no naka ni chousen no chi ga arun desu ka?

nagareterun desu ka or arimasu ka is ok too.

Here's one way to check but it might not work. Are you sure you and your dad have Japanese citizenship? Koseki? Only Japanese citizens have that and a lot of Korean-Japanese do not, which is why they face discrimination (but it's hard for them to get). There's more and more famous Japanese revealing their Korean ancestry so perhaps the stigma is lifting a bit.

Zainichi (kankokujin) is how to say Korean-Japanese in Japanese. (Sangokujin is the derogatory term I would avoid.)

You could say to your dad, "watashitachi wa zainichi kankokujin desu ka?" (Are we Korean-Japanese?)

Speak clearly and say: Dad, Talk to me about our background, where did we come from.

watashi ni wa kankoku-jin no chi ga majitte iru no? jibun no 'roots' wo shiritain dakedo.(casual)

say, syonara akaue, dattebayo