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The best aspects of proving one parent unfit are the following:
Abuse/NeglectMental Illness or DeficiencyAlcohol or Drug Induced IncapacityAbuse/Neglect or Loss of Rights of Another ChildSexual AbuseFelony assault of child or siblingMurder/Manslaughter of sibling or child
While those above are the best routes the fact is very few parents fall under those guidelines. Proving one parent unfit is never easy. Often the courts require you to not only have proof but prove you are better equipped to care for the child. If the other parent can be proven unfit by other reasons such as he/she is not caring for the child, not providing a stable home the judge will consider this as well. In the end the judge must rule in the best interest of the child.
I wish you the best,