Can I build a French drain with old concrete and brick instead of rock?
These materiasl will work. Lining the trench with landscape fabric and wrapping it over the top (of the rubble) will keep silt from clogging your drain.  For higher volume flow, I'd add a perforated pipe at the bottom, but it's surely overkill for just a sink. 

Definitely make your rubble somewhat uniform, for best support for the overburden (assuming you are burying the drain).