Hi here is my take on this:

A. who have no high school diploma but who have families with a record of crime - 'but' suggests that the second part should introduce some contrast such as, who have no high school diploma but who have high IQ. Also families with a record of crime does not sound correct. Eliminate.

B. without a high school diploma and families having a criminal record - implies without families having a criminal record. Eliminate.

C. without a high school diploma whose families have a record of crime - Just one subordinate clause. Clear and concise.

D. whose families have criminal records and lacking high school diplomas - it's not clear whetherlacking high school diplomas modifies the 'offenders' or the 'families'. Eliminate.

E. lacking high school diplomas and also having families having criminal records - The structure lacking X and also having Y is not okay. Also clustering of "having" is wrong. "having families that have criminal records" could be much better - Eliminate.

~Binit.