Yes, al-Husayn (may Allah be pleased with him) was killed as a martyr.
That was when the people of Iraq (Kufah) wrote to him and asked him to come out to them so that they could swear allegiance to him as their ruler, which happened after the death of Mu‘aawiyah (may Allah be pleased with him), and the accession of his son Yazeed to the caliphate.
Then the people of Kufah turned against al-Husayn after ‘Ubaydullah ibn Ziyaad was appointed as governor of the city by Yazeed ibn Mu‘aawiyah and killed Muslim ibn ‘Uqayl, who was al-Husayn’s envoy to them. The hearts of the people of Iraq were with al-Husayn, but their swords were with ‘Ubaydullah ibn Ziyaad.
al-Husayn went out to them, not knowing of the killing of Muslim ibn ‘Uqayl, or of the people’s changed attitude towards him.
Wise men who loved him had advised him not to go out to Iraq, but he insisted on going out to them. Among those who gave him this advice were: ‘Abdullah ibn ‘Abbaas, ‘Abdullah ibn ‘Umar, Abu Sa‘eed al-Khudri, Jaabir ibn ‘Abdullah, al-Miswar ibn Makhramah, and ‘Abdullah ibn az-Zubayr (may Allah be pleased with them all).
So al-Husayn travelled to Iraq, and halted at Karbala’, where he came to know that the people of Iraq had turned against him. So al-Husayn asked the army that came to fight him for one of three things: either to let him return to Makkah, or to let him go to Yazeed ibn Mu‘aawiyah, or to let him go to the frontier to fight in jihad for the sake of Allah.
But they insisted that he should surrender to them, and al-Husayn refused, so they fought him, and he was killed wrongfully as a martyr (may Allah be pleased with him).
Al-Bidaayah wa’n-Nihaayah (11/473-520
Shaykh al-Islam Ibn Taymiyah (may Allah have mercy on him) said:
Yazeed ibn Mu‘aawiyah was born during the caliphate of ‘Uthmaan ibn ‘Affaan (may Allah be pleased with him) and did not meet the Prophet (blessings and peace of Allah be upon him). He was not one of the Sahaabah, according to scholarly consensus, and he was not one of those who were well known for religious commitment and righteousness. He was one of the Muslim youth, and he was not a disbeliever or a heretic. He became the caliph after his father died, despite the objections of some of the Muslims and with the approval of some of them. He was courageous and generous, and he did not outwardly blatantly commit immoral actions, as some of his opponents said that he did.
During his rule, a number of grievous events occurred, one of which was the killing of al-Husayn (may Allah be pleased with him). Yazeed did not issue orders that al-Husayn be killed, and he did not express joy at his killing. He did not poke the severed head of al-Husayn (may Allah be pleased with him) with a stick, and the head of al-Husayn (may Allah be pleased with him) was not brought to him in Syria, but he did issue instructions that al-Husayn be prevented from achieving his goal, even if that involved fighting him. But those who received his instructions went further than that.
Al-Husayn (may Allah be pleased with him) asked them to let him go to Yazeed or let him go to the border and keep watch there, or let him go back to Makkah, but they insisted on taking him prisoner and ‘Umar ibn Sa‘d issued orders to fight him, and they killed him wrongfully – him and a number of his family members (may Allah be pleased with them). His killing was a major calamity, because the killing of al-Husayn, and of ‘Uthmaan before him, was among the main causes of turmoil in this ummah, and their killers are among the most evil of people before Allah. End quote.
Majmoo‘ al-Fataawa (3/410-413)
He also said (25/302-305):
When al-Husayn ibn ‘Ali (may Allah be pleased with him) was killed on the day of ‘Ashoora’, he was killed by the transgressing, wrongdoing group. Allah honoured al-Husayn with martyrdom, as He honoured other members of his family; He honoured Hamzah and Ja‘far with martyrdom, as well as his father ‘Ali and others. His martyrdom was one of the means by which Allah raised him in status, for he and his brother al-Hasan will be the leaders of the youth among the people of Paradise, and high status is only achieved by means of trials, as the Prophet (blessings and peace of Allah be upon him) said, when he was asked which of the people are most sorely tested? He said: “The Prophets, then the righteous, then the next best and the next best. A man will be tested to a degree commensurate with his level of religious commitment. If there is firmness in his religious commitment, his test will be greater, and if there is any weakness in his religious commitment, the test will be reduced for him. Trials will continue to befall the believer until he walks upon the earth with no sin on him.”
Narrated by at-Tirmidhi and others.
Al-Hasan and al-Husayn had previously been granted high status by Allah, may He be glorified, and they did not go through the same trials and hardships that had befallen their predecessors, because they were born at a time when Islam was prevalent and they had grown up with honour and dignity. The Muslims venerated them and honoured them, and when the Prophet (blessings and peace of Allah be upon him) died, they had not yet reached the age of discernment. The blessing that Allah bestowed upon them was that He tested them with that which caused them to join the other members of their family, just as He tested others who were better than them. ‘Ali ibn Abi Taalib was better than them, and he was killed as a martyr. The killing of al-Husayn was an event that provoked turmoil among the people, just as the killing of ‘Uthmaan was one of the main causes of turmoil among the people, as a result of which the ummah is still divided until the present day.
When al-Husayn (may Allah be pleased with him) went out and saw that things had changed, he asked them to let him go back (to Makkah) or to let him go to one of the border regions, or to let him go and join his paternal cousin Yazeed. But they did not let him do any of these things, unless he surrendered to them and they took him captive. They fought him, so he fought back, then they killed him and a number of those who were with him wrongfully and as martyrs. Allah honoured him with martyrdom and caused him to join the pure and good members of his family, and He humiliated thereby those who wronged him and transgressed against him. End quote.