Imaam al-Hasan ibn 'Ali رضي الله عنهما was the fifth rightly-guided caliph. 

Shaykh Dr. 'Ali Muhammad as-Sallabi حفظه الله said,

"After allegiance was sworn to Ameerul Mu'mineen al-Hasan ibn 'Ali رضي الله عنهما, he remained the caliph of the Hejaz (the Western region of the Arabian Peninsula that includes Makkah and Madinah), Yemen, Iraq and other regions for approximately seven months (with some saying six or eight months). During that period, his caliphate was a Rightly-Guided Caliphate. This is because it was the completion of the era of the Rightly-Guided Caliphate, which the Prophet صلى الله عليه وسلم had said would last for thirty years before turning into a kingship. 

Imam at-Tirmidhi رحمه الله narrated, with a chain going back to Safeenah رضي الله عنه, the freed slave of the Messenger of Allah صلى الله عليه وسلم, that he said: 'The caliphate in my Ummah will last for thirty years; after that, it will become a kingship.' (A good hadith recorded by at-Tirmidhi) 

Imaam lbn Katheer رحمه الله commented on this hadeeth: 'The thirty-year period was only completed with the caliphate of al-Hasan ibn 'Ali رضي الله عنهما. He gave up the caliphate to Mu'awiyah رضي الله عنه in Rabee' I, 41 AH, and that was the completion of thirty years after the death of the Messenger of Allah صلى الله عليه وسلم, for he had died in Rabee' 1, 11 AH. This is one of the signs of his prophethood, may the blessings and peace of Allah be upon him.' 

Thus, al-Hasan رضي الله عنه was the fifth Rightly-Guided Caliph. 

This was also narrated by Imam Ahmad رحمه الله from the hadeeth of Safeenah رضي الله عنه: 'The caliphate will last for thirty years; after that, it will become a kingship.' (Recorded by Ahmad with a good chain of narrators.) 

Imaam Abu Dawood رحمه الله narrated it as follows: 'The caliphate in the footsteps of prophethood will be thirty years; then Allah will give power and authority to whomever He wills, or will give His power and authority to whomever He wills.' (Recorded by Abu Dawood) 

During the thirty years after his death, there were no rulers other than the four caliphs رضي الله عنهم and al-Hasan رضي الله عنه for a brief period. A number of scholars commented on the Prophet's words صلى الله عليه وسلم, 'The caliphate in my Ummah will be for thirty years,' that the few months during which al-Hasan رضي الله عنه ruled, following the death of his father رضي الله عنه, are included in 'the caliphate in the footsteps of prophethood'. The following are some of those comments: 

Imaam Abu Bakr ibn al-' Arabi (may Allah have mercy on him): 'Al-Hasan رضي الله عنه fulfilled the true promise of the Prophet صلى الله عليه وسلم, who said: "The caliphate in my Ummah will be thirty years, and then it will become a kingship." That belonged to Abu Bakr, 'Umar, 'Uthman and 'Ali رضي الله عنهم, and al-Hasan رضي الله عنه had eight months of it, no more and no less. Glory be to the All-Knowing, besides Whom there is no other Lord.' 

Imaam Al-Qadi 'Iyad (may Allah have mercy on him): 'During the thirty years, there were no rulers other than the four Rightly-Guided Caliphs رضي الله عنهم and the few months during which allegiance was sworn to al-Hasan ibn 'Ali رضي الله عنهما. The hadeeth about the caliphate lasting for thirty years refers to "the caliphate in the footsteps of prophethood," as stated in some reports: "The caliphate in the footsteps of prophethood after me will be thirty years, and then it will become a kingship."' (Muslim) 

Imaam Al-Hafidh Ibn Katheer (may Allah have mercy on him): 'The evidence that he was one of the Rightly-Guided Caliphs is the hadeeth which we narrated in Dalaa'il an-Nubuwwah via Safeenah رضي الله عنه, the freed slave of the Messenger of Allah صلى الله عليه وسلم, according to which he said: "The caliphate after me will be for thirty years." The thirty years were only completed by the caliphate of al-Hasan ibn 'Ali رضي الله عنهما.' 

Imaam Ibn Abil-'Izz al-Hanafi (may Allah have mercy on him): 'The caliphate of Abu Bakr رضي الله عنه lasted for two years and three months, the caliphate of 'Umar رضي الله عنه lasted for ten and a half years, the caliphate of 'Uthman رضي الله عنه lasted for twelve years, the caliphate of 'Ali رضي الله عنه lasted for four years and nine months, and the caliphate of al-Hasan رضي الله عنه lasted for six months.' 

Imaam Al-Mannawi (may Allah have mercy on him): He quoted the words of the Prophet صلى الله عليه وسلم: 'This son of mine is a sayyid, and perhaps through him Allah will reconcile two great groups of Muslims.' (Al-Bukhari) Then he said: 'That was when allegiance was sworn to him [al-Hasan رضي الله عنه] after his father رضي الله عنه was gone, and he became the legitimate ruler for a period of six months, which completed the thirty years which the Prophet صلى الله عليه وسلم said would be the length of time the caliphate would last, after which it would become kingship.' 

Imaam Ibn Hajar al-Haythami (may Allah have mercy on him): 'He is the last of the Rightly-Guided Caliphs رضي الله عنهم, according to the statement of his grandfather صلى الله عليه وسلم. He became the caliph after the murder of his father, with the oath of allegiance of the people of Kufa. He remained the caliph for six months and several days. He stayed in Kufa for six months and a few days as a legitimate caliph and a just and sincere ruler, fulfilling what his grandfather صلى الله عليه وسلم had said: "The caliphate after me will be for thirty years." Those six months were the completion of those thirty years. His caliphate is referred to in reports, and there is scholarly consensus concerning it, so there is no doubt about its legitimacy. Ahl as-Sunnah wal-Jama'ah believe that the caliphate of al-Hasan رضي الله عنه was legitimate and that it was the final part of the caliphate of prophethood that the Prophet صلى الله عليه وسلم had foretold would last for thirty years.'" 

Shaykh Dr. Ali M. Sallabi حفظه الله, Al-Hasan Ibn 'Ali, His Life & Times (IIPH), pp. 202-205