John Alcindor was born in Trinidad in 1873. Educated at St. Mary's College he was awarded an Island Scholarship and decided to attend medical school in Edinburgh. Alcindor graduated from the University of Edinburgh in 1899 with a first-class honours degree.
Alcindor moved to London and over the next few years worked at hospitals in Plaistow, Hampstead and Camberwell. In July 1900 Alcindor attended the Pan-African Conference held at Westminster Town Hall. There were 37 delegates from Europe, Africa and the United States. Those attending included Samuel Coleridge Taylor, John Archer, Dadabhai Naoroji, Sylvester Williams and William Du Bois. At the conference a large number of delegates made speeches where they called for governments to introduce legislation that would ensure racially equality. Michael Creighton, the Bishop of London, asked the British government to confer the "benefits of self-government" on "other races as soon as possible".
After the conference the Pan-African Congress wrote to Joseph Chamberlain, the British colonial secretary, suggesting that black people in the British Empire should be granted "true civil and political rights". Chamberlain replied that black people were "totally unfit for representative institutions". Sylvester Williams responded to this by writing to Queen Victoria about the system "whereby black men, women, and children were placed in legalized bondage to white colonists". The letter was passed to Chamberlain who replied that the government would not "overlook the interests and welfare of the native races."
In 1907 Alcindor established his own medical practice in Paddington. He also carried out research and he published articles on cancer, tuberculosis and influenza in the British Medical Journal and the General Practitioner. He pointed out that his research suggested that poverty, low quality food and unbalanced diets played an important role in poor health.
In 1911 Alcindor married Minnie Martin and the couple had three sons (John, Cyril and Roland). As well as running a medical practice at 37 Westbourne Park Road he worked as Medical Officer of Health for the Paddington Poor Law Guardians.
Alcindor remained active in the struggle for equal rights and in 1921 he succeeded John Archer as president of the African Progress Union.