Essentially the King of England was a member of the Plantagenet dynasty which came from the French power centre of Anjou. Following 1066 the Normans were eclipsed by the Anjouvites who founded the Angevin Empire in the 12th century. During the 12th century the Plantagenets controlled most of France and England, as you can see from this map here.
The major antagonist of the Plantagenets for mastery of France were the the Capetians who were a powerful family with estates at Paris, and were one of the leading families in the squabble for leadership of the Carolingian Empire, which as we all know, was the successor state of the Merovingian dynasty, itself the Frankish successor of the Roman Empire. King Louis IX Capetian (the Saint- he participated in the 7th and 8th Crusades) during the 13th century re-conquered and married his way into most of the Plantagenet holdings except a small rump in the Duchy of Gascony or Aquitaine (Bordeaux, producer of the valuable English claret wine). The Duke of Aquitaine (later Prince of Aquitaine) was a title handed to the scion of the Plantagenet dynasty (similar to the Prince of Wales being the heir to the throne today), and it was in this capacity that the Plantagenets asserted their claim to leadership in France.
The 100 years war, which has already been mentioned, was an effort by the declining Plantagenets to reassert themselves as the rulers of France. The Duke/Prince of Aquitaine (ie, the future or reigning King of England), was beholden to the King of France (the head Capetian), and was thus a leading noble in the French court system.
As you can see here, as a result of the outcome of the first phase of the 100 years war (after the capture of King John II of France at the Battle of Pointers by the Black Prince-- that is, Prince of Aquitaine), the Treaty of Bretigny arranged for the return of much of the Aquitaine lands to the King of England.
Henry V of Lancaster (one of the other houses of the Plantagenet dynasty) was then able to succeed through his 1420 campaigns to finalize this goal of the return to the Anjouvites of the Kingdom of France by securing the Treaty of Troyes which recognized that he would inherit the Kingdom of France following the death of the ruling Capetian (now under the leadership of the house of Valois).
This of course didn't happen because the Capetians were able to reverse their misfortune thanks to the exploitation of the nationalistic religious fanatic Joan of Arc to break the Siege of Orleans. By the middle of the 15th century Charles VII Valois was powerful enough to declare the Plantagenet holdings forfeit (except for Calais). This caused a civil war in England whereby the House of Tudor (Lancaster-Plantagenets) defeated the York-Plantagenets.
The Tudors were finally toppled when James I of Scotland became King of England following the death of Elizabeth I, the last of the Plantagenets.
The rulership of France continued to pass between the various Capetian dynastic sub-houses, notably the Orleans, Bourbons and Valois until the French Revolution blew everything up.