Son of William Miller of Glenlee (d. 1753) and Janet Hamilton. In 1753 he married Margaret Murdoch, a daughter of John Murdoch of Rosebank, Provost of Glasgow. He later (1768) married Anne Lockhart, daughter of John Lockhart of Castlehill.1
Miller was called to the Scottish Bar on February 21, 1742, and in 1755 he became solicitor of Excise. In 1759 he was appointed Solicitor General for Scotland, and in 1760 became Lord Advocate. Appointed Lord Chief Justice of the Court of Session with the title of Lord Barskimming in 1766, a title he later changed to Lord Glenlee. Lord President of the Court of Session from 1788 until his death.
From 1761 to 1766 he served as MP for Dumfries Burghs. His estate was next to Ballochmyle, near Mauchline, and he was thus almost a neighbour of the Boswells at Auchinleck.
He was an early patron of poet Robert Burns, who referred to him as the aged judge in The Vision I.XX.2
Boswell spent the evening with Miller, James Macdonald, Captain Johnstone and Andrew Erskine at Lord Eglinton's on January 5, 1763, in London. On March 17, 1763, they dined together. On this occasion Boswell didn't like "Mrs. Miller's abominable Glasgow tongue", and "resolved never again to dine where a Scotchwoman from the West was allowed to feed with us". However, as Pottle remarked almost 200 years on, "he later married one".1
Shortly before leaving for Holland in July 1763, Boswell wrote to Miller "in the easy style of a companion, as he and I were always easy, and mention[ed] with satisfaction my having more rational views". Boswell was referring to his decision to continue his law studies to become an advocate.2
In 1786 Boswell and Miller were involved in a dispute - with each other - "over the stool [...] of Turnerhill wood, which Miller's uncle, then the owner, had exchanged with Lord Auchinleck for other land about forty years earlier."3 In 1789 Boswell was involved, together with Miller and Keith Stewart, in a dispute over the relocation of a toll-bar in the parish of Muirkirk.4