Briscoe was born in Wrexham, Wales and educated at Ruthin School.
He then studied at Jesus College, Oxford, gaining a first class Bachelor of Arts degree in Literae Humaniores in 1833.
He was ordained priest in 1837 and became curate of Henllan, Denbighshire, holding this post until 1840.
In 1843, he was awarded a Bachelor of Divinity degree by the University of Oxford, obtaining his doctorate in 1868.
[1] Briscoe's scholastic interests encompassed ancient and modern languages, particularly Welsh.