Melnik is the smallest town in Bulgaria and as such does not offer the nightlife you would see in big cities :)
So, the best you could do is to get out for a walk, check out the taverns on your way, pick one and have a lot of wine + some food.
I asure you that you will not have problems finding taverns. In fact my impression was that there were more taverns than local people around :)))
Dress Code: You are a tourist, what ever you would feel comfortable :)))
Nightlife is located along the melnishka dol, Coming from the busstation, there are winebars located on the left side of the mainstreet, just before the second bridge over the river. These have outdoor terracaces and a bottle of plonk is 5 lv. On the other side of the river, when you cross the bridge, there are terraces also, such as the garden of Loznitsite tavern, although the staff was not so friendly over there.