Even today, there's still a stigma attached to clinical depression -- so I imagine that depression could have arguably been viewed as a lack of will or mental strength, rather than an actual mental illness and imbalance.
Today, despite the stigma, there are obviously more people seeking to get diagnosed and treatment; is there any indication that people avoided seeking treatment, or perhaps even doctors refused to acknowledge that it was something to be treated?
I don't know the answer, but I think I can point you in the right direction. Andrew Solomon, in his book "The Noonday Demon" (wich I higly recommend) wrote that, contrary to popular belief, depression it is not a contemporary problem, exclusively. Different times and different cultures register the existence of clinical depression - with different names, but with similar symptoms. The book has a chapter in wich this question is discussed. Before you read the book, you can check here: http://www.bookrags.com/studyguide-noonday-demon/chapanal008.html
Intriguingly, we do have evidence of a Hittite ritual of all things possibly dealing with depression. CTH 432 is a Hittite ritual commonly known as a ritual treating insomnia, but Beckman argues that the symptoms described more closely reflect depression. Interestingly, considering our contemporary attitude towards depression, the illness is treated precisely in accordance with contemporary Assyro-Babylonian conventions under Hittite influence. The patient is asked to atone for any lapses or sins which they may have been responsible for, the priest makes offerings to purify the patient, the ceremonies are preformed within circles outlined of flour as is the normal way of creating a ritual space in Mesopotamian cultic practice(c.f. the "Drawing a 'house of the dream-god' in Gilgamesh),and torches of colored wool are burned(an act found in other Hittite texts describing purification or release from evil). So in many ways the attitude towards depression parallels ours; depression is not a failure of will but an external evil inflicted like other misfortunes or illnesses from the gods and best treated not by moral adages but by a ritual specialist who is able to correctly counteract the evil.
This is borne out from other medical texts that appear to concern depression; in these texts, disorders which appear to exhibit symptoms typical of depression are grouped alongside epilepsy as diseases that can be inflicted by the 'hand of a ghost' and are subject to similar procedures and in some cases similar disorders are described in diagnostic handbooks for exorcists as potential results of sorcery, and likewise to be counteracted by magico-medical means. So we do have evidence of psychological disorders being recognized as a distinct category of medical problems and treated accordingly. It would incidentally be interesting to see if any of these remedies helped the disorders even simply by providing psychological consolation or allowing the patient to articulate and process his or her emotions, although such a proposal is probably untestable without serious violation of medical ethics.