Well, after the English-Scottish-British monarch list, I've decided to show the Frankish-French list for comparison. Underlined names show the monarch in question was also an emperor.
Kings of the Franks (Note:This is only the main ruler of the Franks, which is why the numbers will occasionally be off)
428-448: Chlodio Longhair, son of Theudemeres, King of the Salian Franks (House of the Merovingians)
448-457: Merovech, King of the Salian Franks (House of the Merovingians)
457-481: Childeric I, King of the Salian Franks (House of the Merovingians)
481-511: Clovis I (House of the Merovingians)
511-558: Childebert I (House of Merovingians) (King of Paris)
558-561: Chlothar I, the Old (House of the Merovingians)
561-561: Charibert I (House of the Merovingians) (King of Paris)
567-584: Chilperic I (House of the Merovingians) (King of Paris and Neustria)
584-629: Chlothar II, the Great, the Young (House of the Merovingians)
629-639: Dagobert I (House of the Merovingians)
639-657: Clovis II, the Lazy (House of the Merovingians) (King of Neustria and Burgundy)
657-673: Chlothar III (House of the Merovingians)
673-675: Childeric II (House of the Merovingians)
675-691: Theuderic III/Thierry III (House of the Merovingians)
691-695: Clovis IV (House of the Merovingians)
695-711: Childebert III, the Just (House of the Merovingians)
711-715: Dagobert III (House of the Merovingians)
715-721: Chilperic II (House of the Merovingians)
721-737: Theuderic IV/Thierry IV (House of the Merovingians)
737-741: Charles, the Hammer, Martel (House of the Carolingians) (Mayor of the Palace)
741-743: Pepin, the Short (House of the Carolingians) (Mayor of the Palace)
743-751: Childeric III (House of the Merovingians) (Restored and Deposed by Mayor Pepin, last of the Merovingians)
751-768: Pepin, the Short (House of the Carolingians)
768-771: Carloman I (House of the Carolingians) (Co-Ruled with his brother, Charles)
768-814: Charles I, the Great, Carolus Magnus, Charlemagne (House of the Carolingians)
814-840: Louis I, the Pious, the Debonaire (House of the Carolingians)
Kings of the West Franks
840-877: Charles II, the Bald (House of the Carolingians)
877-879: Louis II, the Stammerer (House of the Carolingians)
879-882: Louis III (House of the Carolingians) (Co-Ruled with his brother, Carloman)
879-884: Carloman II (House of the Carolingians) (Succeeded by his cousin, Charles, son of Louis the German)
884-888: Charles, the Fat (House of the Carolingians) (Charles the Fat is not numbered)
888-898: Odo/Eudes of Paris (House of the Robertians) (Odo was elected King as no one wanted Louis II's son Charles to be king. Son of Robert the Strong)
898-922: Charles III, the Simple (House of the Carolingians) (The Aforementioned son of Louis)
922-923: Robert I (House of the Robertians) (Brother of Odo)
923-936: Rudolph/Raoul (House of the Bosonids) (Seriously. There was once a King Rudolph. Son-in-law of Robert I)
936-954: Louis IV, from Overseas, d'Outremer (House of the Carolingians) (Son of Charles the Simple)
954-986: Lothair (House of the Carolingians)
986-987: Louis V, the Lazy, the Indolent, the Sluggard, Who did Nothing (House of the Carolingians) (Last of the line of Charles the Bald)
Kings of France
987-996: Hugh, Capet (House of the Robertians/House of Capet)
996-1031: Robert II, the Pious, the Wise (House of Capet)
1017-1025: Hugh Magnus (House of Capet) (Co-Ruler with his father Robert II, never ruled in his own right)
1031-1060: Henry I (House of Capet)
1060-1108: Philip I, l'Amoreux (House of Capet)
1108-1137: Louis VI, the Fat (House of Capet)
1129-1131: Philip of France (House of Capet) (Co-Ruler with his father Louis VI, never ruled in his own right)
1137-1180: Louis VII, the Young (House of Capet)
1180-1223: Philip II, Augustus (House of Capet)
1223-1226: Louis VIII, the Lion (House of Capet)
1226-1270: Louis IX, the Saint (House of Capet)
1270-1285: Philip III, the Bold (House of Capet)
1285-1314: Philip IV, the Fair (House of Capet) (King of Navarre by marriage)
1314-1316: Louis X, the Quarreller (House of Capet) (Navarre in Personal Union with France)
1316: John I, the Posthumous (House of Capet)
1316-1322: Philip V, the Tall (House of Capet)
1322-1328: Charles IV, the Fair (House of Capet) (Last of the Direct Capetians. Upon his death the union between France and Navarre ended. Crown disputed between Philip of Valois and Edward III, King of England)
1328-1350: Philip VI, of Valois, the Fortunate (House of Capet-Valois)
1350-1364: John II, the Good (House of Valois)
1364-1380: Charles V, the Wise (House of Valois)
1380-1422: Charles VI, the Beloved, the Mad (House of Valois)
1415-1422: Henry, King of England (House of Lancaster) (Named King of France in peace treaty with Charles VI, not recognized)
1422-1461: Charles VII, the Victorious, the Well-Served (House of Valois)
1461-1483: Louis XI, the Prudent, the Universal Spider (House of Valois)
1483-1498: Charles VIII, the Affable (House of Valois)
1498-1515: Louis XII, the Father of the People (House of Capet-Orleans)
1515-1547: Francis I, the Father and Restorer of Letters (House of Capet-Angouleme)
1547-1559: Henry II (House of Capet-Angouleme)
1559-1560: Francis II (House of Capet-Angouleme) (also King-Consort of Scotland)
1560-1574: Charles IX (House of Capet-Angouleme)
1574-1589: Henry III (House of Capet-Angouleme) (also King of Poland ... somehow)
1589-1610: Henry IV, Good King Henri, the Green Gallant (House of Capet-Bourbon) (returned Navarre to Personal Union with France)
1610-1643: Louis XIII, the Just (House of Bourbon)
1643-1715: Louis XIV, Louis le Grand, the Sun King (House of Bourbon)
1715-1774: Louis XV, the Beloved (House of Bourbon)
1774-1792: Louis XVI, Louis the Last (House of Bourbon) (Executed by the Republicans for treason)
1792-1795: Louis XVII (House of Bourbon) (Uncrowned and never ruled)
1792-1804: The First French Republic
1804-1814: Napoleon I, the Corsican, the Great (House of Bonaparte)
1814-1815: Louis XVIII (House of Bourbon) (Brother of Louis XVI)
1815: Napoleon I (House of Bonaparte) (restored for 100 days)
1815: Napoleon II, Franz of Reichstag (House of Bonaparte) (15 day reign not recognized)
1815-1824: Louis XVIII (House of Bourbon) (Restored)
1824-1830: Charles X (House of Bourbon) (Abdicated)
1830: Louis XIX (House of Bourbon) (Abdicated after a grand reign of 20 minutes)
1830: Henri d'Artois (House of Bourbon) (Considered by Legitimists to have ruled as Henry V for a week, not recognized by anyone else)
1830-1848: Louis-Philip I, the Citizen King, the Pear-Faced, the July Monarch (House of Capet-Bourbon-Orleans) (Abdicated, Orleanists consider his grandson Philippe, Comte d'Paris to be either Louis-Philip II or Philip VI)
1848-1852: The Second French Republic
1852-1870: Napoleon III, Louis-Napoleon (House of Bonaparte) (Abdicated)
1870: Napoleon IV (House of Bonaparte) (Never crowned or ruled, only recognized by Bonapartists)
1870-1871: The Government of National Defense (The Paris Commune)
1871-1940: The Third French Republic
1940-1944: The French State (not recognized)
1940-1944: Free France
1944-1946: Provisional Government of the French Republic
1946-1958: The Fourth French Republic
1958-Present: The Fifth French Republic
Modern Claimants to the throne of France:
The Legitimists: Refuse to recognize the July Monarchy and all subsequent governments. Union Legitimists refuse to recognize Henri d'Artois, the Comte de Chambourd's naming of Philippe of Paris as his heir. Current Legitimist Claimant is Louis Alphonse, Duke of Anjou, who would be Louis XX (House of Capet-Bourbon-Anjou)
The Orleanists: Refuse to recognize the Second French Republic and all subsequent governments. Current Orleanist Claimant is Henri, Comte de Paris et Duc de France, who would be Henry VII (House of Capet-Bourbon-Orleans)
The Bonapartists: Refuse to recognize the Paris Commune and all subsequent governments, as well as the Restoration governments. The Bonapartist claimant is either Charles Napoleon Bonaparte, who would be Napoleon VII (House of Bonaparte), or his son Jean-Christophe Napoleon Bonaparte, who claims the same number.
The Jacobites: Yes, the Jacobites. Refusing to recognize the Hannoverians means they don't recognize George III's renunciation of the English crown's claim to France.