Aldrich was educated at Westminster School under Dr Richard Busby. In 1662, he entered Christ Church, Oxford, and in 1689 was made Dean in succession to the Roman Catholic John Massey, who had fled to the Continent. In 1692, he became Vice-Chancellor of the University of Oxford until 1695. In 1702, he was appointed Rector of Wem in Shropshire, but continued to reside at Oxford, where he died on 14 December 1710. He was buried in Christ Church Cathedral without any memorial, at his own request.
Henry Aldrich was a man of unusually varied gifts. A classical scholar of fair merits, he is best known as the author of a little book on logic (Artis Logicæ Compendium). Although not innovative in the field of Logic itself (it closely follows Peter of Spain's Summulae Logicales), its insistent use by generations of Oxford students has shown it to be of great synthetic and didactic value: the Compendium continued to be read at Oxford (in Mansel's revised edition) till long past the middle of the 19th century.
Aldrich also composed a number of anthems and church services of high merit, and adapted much of the music of Palestrina and Carissimi to English words with great skill and judgment. To him we owe the well-known catch, "Hark, the bonny Christ Church bells."
Evidence of his skill as an architect may be seen in the church and campanile of All Saints Church, Oxford, and in three sides of the so-called Peckwater Quadrangle of Christ Church, which were erected after his designs. He bore a great reputation for conviviality', and wrote a humorous Latin version of the popular ballad A soldier and a sailor, A tinker and a tailor, etc.
Another specimen of his wit is furnished by the following epigram of the five reasons for drinking: