Can an individual kill a homosexual man or should he report him to the government or concerned authorities?

Praise be to Allah, the Lord of the World; and may His blessings and peace be upon our Prophet Muhammad and upon all his Family and Companions.

Without a doubt a Muslim who witnessed such a great evil should do his best to change it according to his ability, since Allah. The Great and Almighty, destroyed the nation of Lut (Alaihi as-Sallam) for such a crime:
{So when Our Commandment came, We turned (the towns of Sodom in Palestine) upside down, and rained on them stones of baked clay, in a well-arranged manner one after another;} [11:82].
In the same vein, the Prophet (Sallallahu Alaihi wa Sallam) said: "If one of you sees Munkar committed (something evil), one has to change it" . [ Muslim ]
But, it is completely wrong to behave as you mentioned in your question, since killing the one who deserves that, establishing the guilt and imposing punishments stipulated in Sharia are among the duties of those who have the authority, not individuals. Otherwise, such a suggestion of yours to kill such people leads to confusion and corruption.
In this concern, Sheikh al-Islam Ibn Taymiyah said: 'No one is allowed to change al-Munkar through committing what is worse than it. For example, if a person assumes the duty of cutting off the thief's hand, lashing one who drinks wine, establishing punishment stipulated in Shari'a, this will cause confusion and corruption, since this will encourage everyone to claim that someone else must be punished. As a rule, such a duty is confined to rulers'.
But, if al-Munkar exists and being done, a Muslim has to change such al-Munkar provided that it does not lead to a greater evil or disorder. In other words, one who assumes changing al-Munkar should not exceed what is needed to achieve his goal.
On the other hand, Imam al-Haramain said: 'Individuals are allowed to deter one from committing a major sin forcibly if he is not restrained from it by words. But, if such a situation may lead to fight and using arms, then taking an action against wrongdoers should be confined to rulers only'.
al-Ghazzali , in his book al-Ihyaa, stated the following comprehensive rule: 'Individuals are only to object to al-Munkar. Other means and choices such as punishment for previous crime or restraining from eventual felony is the duty of a ruler, not the subject'.
Here, we state that the best thing to do is to inform rulers and those in authority so they can put Allah's rulings and orders into practice; otherwise, it is better to give advice, to urge a wrongdoer to avoid such deeds and then to set the matter right according to one's ability.
Allah knows best.