I want to do some research on Thai historical figures, specifically the Aphaiwong Family, and wondered if anybody could recommend which libraries might be best for me to visit in Bangkok. I can read English and Chinese but only just starting out with Thai.

Any advice would be most appreciated.

http://www.siam-society.org/library/li_index.html

maybe neilson hays as it's an older collection, check university libraries, 
e.g. chulalongkorn and thamasat though you'd contact them first.

I think you could try to contact the Abhaiwongse family to
see if their archive is deposited in an institution;
many of these families keep on to their archives themselves.

I think Thammasat University is renowned also for historical studies. They have two campuses in Bangkok: Tha Prachan and Rangsit. Try getting access to their libraries.

Map as Libraries in Bangkok

Thank you for all of these very helpful answers.

I will start planning my trip to BKK.