His first cousin was Cook Islands politician Geoffrey Henry.
Henry became the Cook Islands' first Premier in 1965 after having lived in New Zealand for several years before returning to his home country. Before the country's accession to a self-governing status, the country had been ruled by New Zealand since annexation in 1901. Global changes to post-colonial attitudes regarding nations colonized by colonial powers prompted New Zealand to alter its political course as regards to the Cook Islands, Niue and Tokelau. The New Zealand government wanted to give its colonial territories independence but, prior to 1965, the Cook Islands had no political system. The general population rarely took any interest as to what the country's future should be. Although Albert Henry was a resident of New Zealand, his interest to become politically involved in the Cook Islands was partially driven by those linked to the New Zealand government at the time. Over the years, the New Zealand government have wanted to let go of the Cook Islands from its administration but the Cook Islands and its people had never shown interest in severing this relationship.
In the early 1960s, the New Zealand government had offered the Cook Islands four options concerning the future of the islands. They were (a) complete independence, (b) assimilation within New Zealand, (c) self-government while remaining associated with New Zealand via citizenship, and (d) membership in a future Polynesian federation. The third option was chosen by the Legislative Assembly of the Cook Islands, and the 1965 election was planned as an intervening election before the arrangements for self-government were finalised. Henry's CIP supported the third option; it campaigned as a party that would implement self-government while maintaining New Zealand citizenship for Cook Islanders.
Henry was not eligible to run for election to the Legislative Assembly in the 1965 Cook Islands election because he had not been a resident of the Cook Islands for three years. The CIP had Henry's sister, Marguerite Story, run in the Te-au-o-Tonga riding as a "stand-in" for Henry. After the CIP formed the government, it quickly changed the residency requirement from three years to three months, and Story resigned the seat so that Henry could run in the by-election. Henry won the by-election in Te-au-o-Tonga and shortly afterwards he was selected as the first Premier of the Cook Islands. (In return, Henry ensured that Story was elected Speaker of the Legislative Assembly.) Henry was re-elected consistently in the election of 1968, 1972, 1974 and March 1978. Upon being elected Premier, Henry did much to unify the Cook Islands and to promote its newly awarded self-government. His government quickly approved the proposed constitution, which awarded self-rule to the Cook Islands while maintaining New Zealand citizenship for its residents. He was a charismatic orator in both Māori and in English.
In 1973 Henry introduced a new national flag for the Cook Islands, but the flag was replaced in 1979 after he resigned. He initiated the creation of the House of Ariki. Other achievements include enlarging Rarotonga's airport, building the Rarotongan Resort (now the Rarotongan Beach Resort & Spa), and helping to develop the Cook Islands into a significant international tourist destination. In 1966 Henry introduced a Universal Old Age Pension Scheme in which every person 65 years and over received a small government-sponsored pension.
During the 1978 elections, Henry had become embroiled in a scandal involving overseas Cook Island voters. The 1978 election was the first in which Cook Islanders that lived in New Zealand were not permitted to vote in the Cook Islands election; The CIP had flown hundreds of its supporters from New Zealand to the Cook Islands so that they could vote in the election. There was also evidence that the CIP had bankrolled the tickets with revenue from the sale of postage stamps by the Cook Islands Philatelic Bureau.
After the fraud was discovered, the 1978 election was handed to the opposition party, the Democratic Party; later that year Henry was found guilty of electoral fraud, and his knighthood was revoked. On 16 August 1979, Henry pleaded guilty to two charges of conspiracy and one charge of corruption relating to the use of $337,000 of Cook Islands Government money to fly hundreds of supporters from New Zealand to the Cook Islands in order to vote. He was fined the maximum of $1,400, ordered to pay $2,000 in court costs and placed on three years probation; he was also barred from political office for life.
In 1974 Henry was appointed a Knight Commander of the Order of the British Empire. In April 1980 the knighthood was forfeit due to the previous criminal conviction.
In 1924 the New Zealand All Blacks team called at Rarotonga en route to their tour the United Kingdom and France. Wanting a local team to train against, Albert Henry, 17 years old and just returned from New Zealand, was asked to get some boys together. A scratch team was put together and the game was played on the Takamoa Grounds. The game ended in a scoreless tie. Henry was very active in forming rugby clubs, first with Tupapa, later with Arorangi where he was a teacher.
His health having taken a turn for the worse since the electoral scandal that had cost him his job, his knighthood, and much of his reputation, Henry died on 1 January 1981, aged 73. His body was taken around Rarotonga on the back of a pickup truck, and the road was lined with mourners. His grave can be found at the Avarua CICC Church.