(remarks: there is more than one connection leading to Sir Winston Churchill,
however, I did only choose one particular line)

My line to Sir Winston Churchill:

Edward III (Plantagenet), King of England, 1312- 1377 = 2nd cousin 19+21 times removed, etc.
married Philippa of Hainault, 1311-1369

Edmund (Plantagenet) of Langley, Duke of York, 1341-1402 = 3rd cousin 18+20 times removed, etc.
married Isabel of Castile, 1355-1393

Richard (Plantagenet) of Conisburgh, Earl of Cambridge, 1375-1415 = 4th cousin 17+19 times removed, etc.
married Lady Anne Mortimer, 1388-1411

Richard (Plantagenet) of York, 3rd Duke of York, 1411-1460 = 5th cousin 16+18 times removed, etc.
married Lady Cicely Neville, 1415-1495

Edward IV (Plantagenet), King of England, 1442-1483 = 6th cousin 15+17 times removed, etc.
married Elizabeth Woodville, 1437-1492

Elizabeth (Plantagenet) of York, 1466-1503 = 7th cousin 14+16 times removed, etc.
married Henry VII (Tudor), King of England, 1457-1509

Mary Tudor, 1496-1533 = 8th cousin 13+15 times removed, etc.
married Charles Brandon, Duke of Suffolk, 1484-1545

Eleanor Brandon, 1519-1547 = 9th cousin 12+14 times removed, etc.
married Henry Clifford, 2nd Earl of Cumberland, 1517-1569

Margaret Clifford, ?-1596 = 10th cousin 11+13 times removed, etc.
married Henry Stanley, 4th Earl of Derby, ?-1593

William Stanley, 6th Earl of Derby = 11th cousin 10+12 times removed, etc.
married Elizabeth Vere

*James Stanley, 7th Earl of Derby, 1607-1651 = 12th cousin 9+11 times removed, etc.
married ?

**Lady Amelia Ann Sophia Stanley, ?-1702 = 13th cousin 8+10 times removed, etc.
married ?

***Lord Charles Murray, 1st Earl of Dy(u)nmore, 1661-1710 = 14th cousin 7+9 times removed, etc.
married ?

****Lady Anne Murray, 1687-1710 = 15th cousin 6+8 times removed, etc.
married ?

*****Lady Catherine Cochrane, 1709-1786 = 16th cousin 5+7 times removed, etc.
married Alexander Stewart, 6th Earl of Galloway, ?-1773

John Stewart, 7th Earl of Galloway, 1736-1806 = 17th cousin 4+6 times removed, etc.
married Anne Dashwood, ?-1830

George Stewart, 8th Earl of Galloway 8th, 1768-1834 = 18th cousin 3+5 times removed, etc.
married Jane Uxbridge, ?-1842

Jane Stewart, ?-1844 = 19th cousin 2+4 times removed, etc.
married George Spencer-Churchill, 6th Duke of Marlborough, 1793-1857

Spencer-Churchill, John Winston, Duke of Marlborough 7th, 1822-1883 = 20th cousin once +3 times removed
married Vane, Frances Anne Emily, ?-1899

Lord Randolph Henry Spencer-Churchill, Rt.Hon., JP. MP., 1849-1895 = 21st cousin ...
married Jennie Jerome, ?-1921

Sir Winston Leonard Spencer-Churchill, Rt.Hon.MP, 1874-1965 = 22nd cousin ...

Source (by Mark Humphrey):

New England Historic Genealogical Society's NEXUS Vol.13 (1996) pp.166-72 and pp.204-11.
MARLBOROUGH in [Burkes Peerage]

Source (by Leo van de Pas):

*The Complete Peerage, Vol.  IV, page 214 and The Earls of Derby by J.J. Bagley, page 78
**Complete Peerage Vol. I, page 316
***The Complete Peerage, Vol. IV, page 542
****The Complete Peerage, Vol. IV, page 527
*****The Complete Peerage, Vol. V, page 606
(all of these are also in Burke's Peerage, 1938 and 1967 edition)