¿Cuál es tu relación con la lengua inglesa?

¿Cuál es tu relación con la lengua inglesa?
 

Can we say:
What is your relationship with the English language?
It sounds odd to me. Anything more natural.
I want to ask something like if they study English outside the school, because they speak really well, or if their mother is American or British?

Your translation sounds OK, but as you suspect, a little odd. A lot depends on context but maybe I would say:
How did you come to speak English so well?

I agree with Poli. How about

What contact have you had with the English language?

Or..."What was your connection to the English language?"

Or what is your connection to the English language I guess? This one fits the purpose.