Usually the origins of the word "addict" are referred to Latin. Once I read somewhere that such word could have greek roots, from "diké", justice, rights. Something like "adiktoi" could mean "those not entitled to rights", therefore slaves. It makes some sense, at least. Where could we find support to trust or to distrust such claims?
It comes from Latin addico, which means, amongst other things, "to devote", so an addict is someone who is (excessively) devoted to something, from the past participle addictus, "devoted". You can look up the etymology of English words on Etymology Online. Note that the double d could never have come from Greek adikê, nor could the combination -ct.