William Clarke was married twice, first to an unknown woman who bore him two children. 
Children by first marriage to unknown woman
His second marriage was to Katherine Babington Storer, widow, who was from the same line as Anthony Babington. Katherine had several children from a previous marriage to Edward Storer.
Stepchildren -- Katherine's children by her marriage to Edward Storer
Arthur. (Wikidata shows Arthur as William's son. 
Stepchild: Child of Edward Storer and Marie Widmerpole being raised by Katherine as stepmother.
Together, Katherine and William had two more children named John and Martha. 
Children by second marriage to Katherine Babington
All of his sons except John followed him into the trade of apothecary.
1643 English Civil War
During the English Civil War, William sided with the parliamentarians. The town of Grantham was captured by Royalists on 23 March 1643 and on April 11 he was indicted for high treason. On 11 May 1643, however the town was recaptured by Oliver Cromwell and Clarke was released. 
Following the war William attained wealth and landholdings, but lost much of his fortune following the restoration.
1654 Isaac Newton Connection
William Clarke provided lodgings for a young Isaac Newton whilst he attended the King's School in Grantham.
In 1654, William provided boarding to Isaac Newton as he would be attending the King's School with Edward and Arthur Storer. Newton's mother remained in Woolsthorpe-by-Colsterworth, which was about eight miles away from the Clarke residence. Many of Newton's biographers have noted that it was the lessons learned from Clarke that sparked Newton's interest in chemistry.
According to the Clarke's grandson, Ralph Clarke, also an apothecary in Grantham, Newton left signs of his presence in the garret where he slept in the apothecary’s house on Grantham’s High Street: he carved his name into the boards, and drew charcoal drawings of birds and beasts, men and ships, and abstract shapes on the walls. Newton was said to have had 'a passion' for Clarke's step-daughter, Katherine Storer (later Mrs. Vincent).
Clarke eventually retired to Loughborough in Leicestershire, leaving his business to his son, William. William apparently had some tensions with his family as his 1671 will contained a clause stating that: "If any be not thanken (sic) that person or persons shall have none".
"In the name and service of the Lord my God, I William Clarke of Loughboro in ye County of Leicester, apothercary, being at his present position by the mercy of God in good and perfect health of body and also of a sound mind, judgement and understanding, so make and ordaine this my Last Will and Testament in manner and form following.
Placing my soul into the hand of God, my merits, humbly and earnestly beseeching Him to adopt of me only in my merits, meditation and confession of Jesus Christ my only Saviour and Redeemer and humbly beseeching Him to beget and [illegible] in me more and more to that lovely home wherein sometimes I pause, glad because of my salvation in the same Jesus our real friend of the bible, and of never fading importance, received for this reason for all times, now awakened by the power of God, brought by the faith to Salvation. Also I commit my body and soul to be decently buried at the supervision of my friends.
Here I bequest and give my personal estate to be disposed of as followeth.
First I give unto my good and loving wife Katherine Clarke all at my messuage and endourment now in the tenure and occupation of Widow Kirke, butcher, situate in Grantham close by the Wood-Yarde. Also the kitchen belonging to the same, and with all the other appertainences so belonging thereto, to be disposed of to her own use and benefit during her naturall life. *And a further Istate to be disposed of as followeth: And I give and bequeath unto my Sonne Joseph Clarke all that remaining part of my tithe and interest in the shop, with all the wares, goods and the deed books thereunto belonging and remaining part therein. I set him up at Loughboro all I reserved to myself during my naturall life.
All the rest of my Istate, real and personal, lands, houses, goods and chattells whatsoever. My Will is that the they shall be equally and indifferently be divided betwixt my daughter Martha Clarke and my Sonne John Clarke, provided notwithstanding that in both the said hands, various goods and chattels upon equal preferrment shall amount about the double value of the shop, met by goods, wares and deed books thereto belonging.
Upon my wish is that the amount surplus or overplus or double value shall be sub-divided into three parts and distributed equally at my loving brethren Dr Humphrey Babington and my loving brethren Mr Joseph Clarke, Physician, betwixt my sonne Joseph Clarke, daughter Martha Clarke, and my sonne John Clarke at Low Estate, of my said wife Katherine Clarke, as before.
As many or any of them shall be disposed of for their preferrment in marriage of otherwise according to the discretion of my said wife, and my said wife in the meantime to advise them whereof as the occasion may require.
That I give unto my sonne William Clarke from the day and date of my decease, my late dwelling house in Grantham now rent in his name and living and the garden in Deadmans Lane with all the appertainences thereto belonging, though it is not to be his by virtue of his mothers. joynture untill my death.
And revoking and legally making null former Wills by me made, I do make and ordain my dear and loving wife to be my supervisor of this my last Will and Testament, and my loving brother Dr Humphrey Babington, Doctor of Divinity, and my loving brother Mr Joseph Clarke, Physician my supervisors thereof, earnestly intreating them to take the name of all my pleasures upon them to see that this my Will be fulfilled. And for their great pains and kindness therein I give unto each of them ten shillings a man.
In witness required of the said William Clarke, have hereunto set my hand and seal this first day of June Anno Domini 1671.
In the presence of: Edward Storer; Susan Storer, Frances Key
If any be not thanken that person or persons shall have none.