Younger brother of the diplomat Benjamin Keene, the family were close friends of Sir Robert Walpole, British Prime Minister from 1721 to 1742.

This connection helped him secure a series of lucrative positions in his early career but it was his relationship with Walpole's successor Thomas Pelham-Holles, 1st Duke of Newcastle that proved most beneficial.

[3] In addition, Rolfe was election agent for Sir Robert Walpole, British Prime Minister from 1721 to 1742; this connection helped the careers of both Benjamin and Edmund.

[4] Through the influence of Sir Robert Walpole, friend of the family, he was educated at Charterhouse School, and was admitted to Gonville and Caius College, Cambridge, in 1730.

On 22 March 1752 he was consecrated in the chapel of Ely House as bishop of Chester, but he did not resign the mastership of his college until 1754.

While at Chester he rebuilt the episcopal palace; George Grenville, in December 1764, proposed that he should accept a transfer to the archiepiscopal see of Armagh, but Keene held out for the diocese of Ely.