Houston was born in the Sumter District of central South Carolina. His parents, Archibald and Margaret Houston, were farmers who had emigrated to the then British colony from Ireland. He attended the College of New Jersey (later Princeton) After his graduation in 1768 he stayed on as a tutor, becoming a Professor of Mathematics and Natural Philosophy (science) in 1771.

When British forces occupied Princeton in 1776 at the outset of the Revolution the college was closed and the students and professors returned home. Houston then joined with the militia of nearby Somerset County and saw action in the area. He was later elected Captain of one of their companies. When the British withdrew from New Jersey in 1777 and the college reopened, he returned to his teaching post.

He was elected to represent Somerset County in the New Jersey General Assembly in 1777. In 1778 he served on the state’s Committee of Safety. Then from 1779 to 1781 New Jersey sent him as a delegate to the Continental Congress. His work in Congress was largely directed to issues of Finance and supply. He began to study law at this time.

He returned to the college, and was admitted to the bar in 1781. Houston also opened a law office in Trenton. During these years he was also named as clerk of the New Jersey Supreme Court. In 1783, he resigned from the college to devote himself to his legal career. He returned to the Continental Congress in 1784 and 1785.

In 1786, Houston was appointed to a commission to study the defects in the Articles of Confederation which joined the states. He went to the Annapolis Convention to discuss the Problem. Instead of proposing changes to the articles, this Convention called for a full Constitutional Convention. When the United States Constitutional Convention assembled in 1787, he went to Philadelphia as a delegate. Houston only remained at the convention for a week before his failing health caused him to withdraw.