He apparently spent his early life in the city of Bristol, Gloucestershire, England. 
William Markham first married on 22 July 1666 at St.James Dukes Place, London England to Anne "Nan" Wright. 
On 10 April 1681 Governor William Penn appointed his relative, William Markham, to serve as acting governor of the Province of Pennsylvania, while Penn was in England. Soon after William sailed for America and landed in Boston, Massachusetts. He made his way to New York arriving in June 1681 where he showed his credentials and took official control of the Delaware territories, which had also been given to Penn.
On 3 August 1681 William arrived at the only town in the colony at that time, Upland (he renamed it Chester, Pennsylvania). He assembled a governing council that included six Quakers and three other early colonists.
As governor, William was tasked with selecting 10,000 acres for the city of Philadelphia, as well as property in the countryside for Penn’s own use. He also bought land from the Indians along the Delaware River and Pennsbury Manor, and began the discourse with Lord Baltimore over the disputed boundary between Pennsylvania and Maryland.
As Deputy Governor, he was in New Castle in October 1682 to welcome William Penn to Pennsylvania. Upon arriving, Penn relieved him of his duties.
William became a representative for the colony in England and lobbied on its behalf in the boundary dispute with Maryland. He served in various other positions including secretary of the province, secretary to the proprietary, a commissioner to sell lands, and an auditor of accounts. William supported John Blackwell over Thomas Lloyd in their dispute over the governorship.
By 1683 he temporarily fell from court popularity and lost control of the colony. The crown gave official control to Benjamin Fletcher, but William Markham served a second term as acting governor as Fletcher's deputy.
William Markham married secondly in 1684 at St.James Dukes Place, London England to Mrs. Joanna Jobson or Johnson (widow of sea captain Eben Jobson or Johnson). 
"The Bloodless Revolution" took place from 1688-1689 in England which involved the overthrow of the Catholic King James II, who was replaced by his Protestant daughter Mary and her Dutch husband, William of Orange. The Glorious Revolution overthrew the Stuart dynasty with which Penn had his connections.
In 1691 Delaware was separated from Pennsylvania, and William became the acting deputy governor of the new colony. William was the acting governor of Pennsylvania from 1693 to 1699.
Penn was reinstated as official governor in August 1694, but William maintained control of the colony until Penn's return from England in December 1699.
William was a member of the Church of England and tended to favour the interests of minority religious groups in the primarily Quaker colony. He had several disputes with the legislative body and issued his own version of the Frame of Government of Pennsylvania in an attempt to resolve some of this conflict. His version of the Frame gave greater power to the lower house of the legislature, the General Assembly, and greatly weakened his executive power as well as that of the upper house, the Council. Markham was criticised by the surveyor-general of customs for, among other things, allowing pirates to run rampant. Pennsylvania did not have the military capacity to protect the Delaware Bay. Penn also complained of fraudulent financial transactions with William, but still had him appointed, through the deputy governor, register-general of wills in 1703.
William Markham died testate on 12 Jun 1704 in Philadelphia, Pennsylvania. 
"Jr." added to help keep father and son differentiated, and don't know how the family at the time differentiated them.
It is claimed that he used a seal with the coat of arms of the Markhams of Sedgebrook, Nottinghamshire, England from whom descended the Markhams of Ollerton, Nottinghamshire, England. So he was probably the bearer of the noble MARKHAM arms ... seal - Az. on a chief or a demi lion rampant issuant.
Marriages at St.James Dukes Place, London England ...
↑ "American National Biography" v14 by John Arthur Garraty, Mark Christopher Carnes, American Council o f Learned Societies published by Oxford University Press in 1999 p516 Text: Markham was twice married, first to Ann Wright, with whom he had one child and in 1684 to Joanna Johnson, a sea captain's widow with whom he had no children. He apparently spent his early life in the city of Bristol.
↑ Diary of Samuel Pepys for 1666/08/05 /Sunday ... "Nan, at Sir W. Pen’s, lately married to one Markeham, a kinsman of Sir W. Pen’s, a pretty wench she is." note: Pepys lived on Pepys Street which is just one block away from St.James Dukes Place in London, England.
↑ "Lawmaking and Legislators in Pennsylvania: 1710-1756" v2 by Craig W. Horle, Temple University, Center for Public Policy & eidited by Craig W. Horle & publsihed by University of Pennsylvania Press in 1997; p525 Text: William Markham, m1: c1666 [Ann] "Nan" Wright; child: Ann, m: James Brown; m2: c1684 Joanna (Johnson); Offices: PA, deputy governor 1681-82; provincial secretary 1685-91; proprietary secretary.
↑ "The Welcome claimants proved, disproved and doubtful with an account of some of their descendants" v2 of Penn's colony; issue#2 of Publications, "Philadelphia Welcome Society of Pennsylvania"
by George Englert McCracken & Genealogical Pub. Company in 1970 p342 Text: 1683/4, Joanna --, widow of Captain Eben Jobson by whom she had a daughter, Elizabeth Jobson who m1: Edward Robinson of Philadelphia, merchant, who's will was probated 04 Nov 1699; m2: Jacob Regnier of Lincoln 's Inn, barrister-at-law and said Elizabeth (Jobson) (Robinson) Regnier.
↑ "Chronicles of Pennsylvania from the English revolution to the peace of Aix-la-Chapelle, 1688-1748" by Charles Penrose Keith v1 issue#89 of Library of American civilization & published by Patterson & White Company in 1917; p145 Text: William Markham had a daughter Ann, who in January 1699/1700, is spoken of as his only child, and whom he, probably when coming to America in 1681, left in the care of William Penn, for she was seven years one of Penn's household. Late in life, Markham married Joanna, a widow, whose daughter married Jacob Regnier, a lawyer in New York. Joanna Markham had a nephew Theodore Colby.
↑ "The Provincial Councillors of Pennsylvania, Who Held Office Between 1733-1776: And Those Earlier Councillors Who Were Some Time Chief Magistrates of the Province, And Their Descendants" by Charles Penrose Keith & Genealogical Publishing Company Inc. of Baltimore, MD in 1883; p5-6 Text: He married at least twice. The wife who survived him was named Joanna. She was not the mother of Markham's only child. When he married her, she was a widow with one daughter, Elizabeth, who m1: Edward Robinson of Philadelphia, PA, merchant, whose Will was probated 04 Nov 1699 and m2: Jacob Regnier of Lincoln's Inn, barrister-at-law and d.s.p. before 03 Aug 1715. The widow Markham removed to New York City, where she died 04 Oct 1726.
↑ Publications of the Genealogical Society of Pennsylvania v3 in 1906 Genealogy and local history p168 Will of Edward Robinson of Philadelphia, PA, merchant; "ye Honble Wm. Markham, Esq." 20 shillings "to buy him a ring"; "to his Lady Joanna Markham" 20 shillings for a ring.
↑ Collections v26 by New York Historical Society in 1894 p153, 402 Text: Will of Elizabeth Regnier; In the name of God, Amen. I, Elizabeth Regnier, widow and executor of Jacob Regnier, late of New York, deceased, being sick and weak; My mother, Joana Markham," is made residuary legatee.
↑ "Select cases of the Mayor's Court of New York City 1674-1784" edited by Richard Brandon Morris & published by American Historical Association in 1935 on ancestry.com citing v2 American legal records; Library of American civilization; New York State p464 Text: 02 July 1723 - 11 Aug 1725 Daniel Regnier and Johanna Markham, Executors, etc: Ver Jacob Moene
Be it remembered that Daniel Regnier and Joanna Markham, widdow, Executors of the last Will and Testament of Elizabeth Regnier, widow deceased, late executrix of the last Will and Testament of Jacob Regnier, Esq., deceased by Henry Wileman, their attorney, fyled their declaration ...
↑ "Genealogies of Pennsylvania families" p557 published by Genealogical Publishing Company in 1982; William Markham, b: c1635; d: 12 4M (June) 1704 Philadelphia (testate).
↑ Death Notice published in "New York Gazette" on 10 Oct 1726; Text: Joanna Markham, widow of Capt. William Markham, former Lt. Gov. of PA, died in NYC 4 Oct.
Genealogy digest v13-15 p12 by Genealogy Club of America in 1982; William Markman m: Ann Penn; William Markham (1635-1704) m1: 05 Aug 1666 Ann Wright; m2: Joanna Jobson.
"Seventeenth century colonial ancestors of members of the National Society Colonial Dames XVII Century 1915-1975" p166 by Mary Louise Marshall Hutton & published by Edwards Brothers in 1976; William Markham (1615-1650) m: Ann Penn; William Markham (1635-1704).