Does the ruling on wrongfully taking a life apply to someone who killed a person that caused him to lose his livelihood and his children, believing that killing him was justified to preserve his family?

It appears that your killing of the man without lawful justification is a grave crime, for intentional killing of a soul is one of the greatest sins in the sight of Allah. Allah has threatened the perpetrator with Hellfire, His wrath, His curse, and a tremendous torment, unless he repents, believes, and does righteous deeds. Many Quranic verses and prophetic traditions have been reported, prohibiting the killing of a soul without just cause, and considering it among the destructive sins and major transgressions. Furthermore, the hadiths indicate that the killer and the killed will be in the Fire if they meet with their swords, and that the first matter to be judged on the Day of Resurrection will be bloodshed.

You must know that provision is in the hands of Allah alone, and no one can prevent your provision. Your belief that the man you killed was cutting off your provision is a corrupt assumption that must be corrected. For the provision of every creature is upon Allah, and there is no preventer of what Allah gives, and no giver of what He withholds.

You must repent and surrender yourself to the guardians of the blood (the victim's family). They may either forgive, or seek retribution (qisas), or take blood money (diyah). Disposing of the consequences of this sin in this world is easier than in the Hereafter.