I need a History Lesson here. I'm not sure if this belongs in this section or the Political section. Mods, please move as you deem appropriate.
I know that at one time Ukraine and Russia were one country, and Kiev was the capitol.
1) When did the country split into two?
It didn't:the central principality of the Rus, Kiev, had declined with the decline (and sack) of Constantinople, its main trading, cultural etc. partner. The end came as capital in 1240 when the Mongols destroyed the city. But the state had always been an appanage state, in which local princes rotated (by succession or sword) on the various thrones of the realm, including that of the Grand Prince of Kiev. Two former Grand Princes sat on the throne after Kiev's sacking (princes were often driven from, and returned to, the various thrones, including Kiev).
Taking 1240 as a conventional date:
St. Maximus, Metropolitan of Kiev and All Rus (1285–1305) left Kiev for Vladimir in 1299, transferring the see there while keeping the metropolitan title. His predecessor Petro Akerovych (1241–1246) took part in the union scheme at Lyons, and for that reason perhaps doesn't appear on some lists of primates of Kiev (I don't know anything on an intervening Cyril III (1247-1281)) This line moved to Moscow with St. Peter in 1325 and became autocephalous in 1442 when Constantinople appostacized and the Russians threw out the apostate Isodore that Constantinople had sent and who had tried to impose the council of Florence on the Orthodox Rus. It became the Patriarchate of Moscow and All Russia in 1589. This all had paralleled secular developments:Vladimir II Monomach Grand Prince of Kiev (1113-1125) had consolidated his power in the principality of Rostov-Suzdal-Vladimir (uniting them, moving his capital to Suzdal from Rostov, and (re)founding Vladimir) as his patrimony before becoming Grand Prince of Kiev, and (according to legend) acquiring the crown Monomokh's cap from his grandfather, the Emperor in Constantinople. (his son Mstislav,the last ruler of united Rus, succeeded him, and upon his death, as the chronicler put it, "the land of Rus was torn apart"). His son George (1099-1157), also in turn Prince of his father's lands and Grand Prince of Kiev, moved the capital to Vladimir, and fortified and founded Moscow as a city. His son, St. Andrew the Pious, actually sacked Kiev in 1169, and instead of assuming the title of Grand Prince there, installed his brother there as his vassal, the throne there going back and forth with claimants from Halych-Volhynia. These dynastic struggles continued until the Mongols came, who appointed Princes of Vladimir who, however, did not move to the capital but stayed in their own patrimony's capital, any of the 11 which George's descendants had carved out of the orinal land of Vladimir Monomokh. St. Mikhail of Tver broke with the Mongols, adopted the titleTsar, and took over Vladimir in 1304. Mikhail had unfortunately nominated someone else as Metropolitan of Kiev, and when St. Peter, nominated by the ruler of Halych-Volhynia, was consecrated by Constantinople, St. Peter threw his support behind Yuriy of Moscow, Michael's second cousin. Ivan III of Moscow, the great-great-great-grandson of Yuriy's brother Ivan I, Grand Prince of Vladimir and Moscow, married into the (defunct) imperial family of Constantinople and took the title of Tsar. In 1547 his grandson Ivan III Grand Prince/Duke (the title is the same as that of Kiev in Slavonic) of Moscow was recognized as Tsar "king/emperor" of all the Rus by Constantinople.
For a score card:http://en.wikipedia.org/wiki/Rulers_of_Russia_family_tree
In 1136 Novogorod, whence the dynasty of Rurik that ruled Kievan Rus first came to rule, dismissed its prince sent by the Rurikids from Kiev. Thereafter the city arrogated to itself the right to choose its prince. Not conquered by the Mongols, it none the less paid tribute to them through Moscow. Dependent on Moscow for grain to feed Novgorod's population, Moscow annexed the city in 1478, putting an end to its practice of choosing princes from the Rurikids, and incorporating it in the Rus led by Moscow.
The Rurikids did not only survive in Moscow: in Halych (now the core of the hot bed of Ukrainian nationalism, in the North West) two lines ruled in succession. The first sprang from Vladimir, the eldest son of Yaroslav I the Wise Grand Prince of Kiev, who ruled in Novgorod but predeceased his father, hence his son Rostislav lost his patrimony of Rostov and became landless until his uncles, all Princes of Kiev in succession, installed him in Halych/Galicia. Roman, son of Mystislav II of Kiev and Agnes of Poland, ruled Novogorod and then inherited Volodymyr-Volynsky/Volhynia, coming from a line that had sat on the thrones of Rostov, Novgorod, Volyn and Kiev, and, when Vladimir's line in Halych died out, took over Halych, uniting it into Halych-Volhynia in 1199. In 1202 he took Kiev and became Grand Prince of Kiev. At his death in 1205, at the hands of his former Polish and Czech allies, his lands became a vacuum into which the Poles, Lithuanians, Hungarians and Czechs came in, until his son Danylo united Halych-Volhynia and Kiev once again in 1239, only to lose Kiev in its sacking, to be succeeded there by a former Grand Prince, his brother in law Michael of Chernigov (himself ruler in succession, among others, of Novgorod and Halych besides Kiev, and a member not only of the Rurikids of Rus' but also the Piasts of Poland (being Danylo's second cousin through Agnes of Poland), the Árpáds of Hungary, the P?emyslids of Bohemia and Moravia), who was martyred by the Tartars in the East (he refused to worship Genghis Khan). Yaroslav III of Novgorod then returned (he had first left Novgorod on the advice of Danylo to go to the throne of Kiev in 1236) to resume the title of Grand Prince of Kiev. In the meantime Danylo had secured the crown as king of Halych-Volhynia from the pope of Rome, but the ruins of Kiev remained in Mongol hands. In Latin his line claimed to be "Rex Rusie [sic]", and in Latin the term "Ruthenia" becomes interchangeable with "Rusia" hence Ruthenia. The claim is made that Constantinople established a second Metropolitan of Kiev in Halych in 1303, but the reports are contradictory. Danylo founded the city Lviv and named it after his son Lev, who moved the capital to there. Another son, John/Svarn, married the daughter of Lithuania's first (and only) king Mindaugas. When Mindaugas' son and successor Vaišvilkas, baptized into the Orthodox Church, resumed the monastic life, he turned the Duchy of Lithunia over to John. Traidenis, a pagan usurper (whose brothers, however, were Orthodox), overthrew and killed John and dragged Lithuania back into paganism (following Mindaugas, who seems to have reverted-if he ever left-paganism after receiving the crown and baptism from the pope of Rome). In the meantime, Lev's son Yurij/George I succeeded him, and his sons Andrew (ruling from Volodymyr in Volhynia) and Lev II (ruling from Galicia) fought off the Tartars and Lithuanians, until falling in battle in 1323. Their nephew Boleslaw-Yuri II of the Piasts was chosen to succeed them-if he embraced Orthodoxy-but poisoned by Orthodox boyars when they suspected he went back to the Vatican, starting the Galicia-Volhynia Wars. Boleslaw had been betrothed to the daughter of Gediminas, the King (or Duke, the title was disputed) of Lithuania (he married another daughter off to the Rurikids in Moscow), another son-in-law of Gediminas Casimir III Piast of Poland, died as the last king of that line, succeeded by his nephew King Louis of Hungary, and in turn by Louis' daughter Jedwiga. Poland made a proposal to Jogaila of Lithuania, whose Russian Orthodox mother, Uliana of Tver and Halych, had intended him to marry Sophia of Moscow. With the marriage of Jogaila and Jedwiga, the Commonwealth of Poland-Lithuania was born, to Halych-Volhynia's detriment. By the time dust of the Galicia-Volhynia Wars dust settled in 1392, Poland got Galicia/Halych and Lithuania had gotten Volhynia. Galicia was never in union with Kiev again until Stalin annexed it to Soviet Ukraine.
The fate of the city of Polotsk and Belarus, not part of Ukraine, relates to your question. Polotsk was ruled by its own Varangian dynasty, who shared the same Nordic origins as the Rurikids, and formed the union of the Slavic Krivichi. When its ruler Rogvolod gave his daughter Rogneda to the rival of the future St. Vladmir, Vladimir sacked the city and took the princess by force. When he divorced her on baptism to take the imperial princess Anna of Constantinople, she returned to Polotsk with her and Vladimir's son Izyaslau as ruler (she herself went into the convent with the name Anastasia). Since Izyaslau predeceased his father, the House of Polotsk lost rights to succeed at Kiev. His son Bryachislav and his descendant made up by asserting their autonomy from Kiev, colonizing Latvia and Lithuania, and using Lithuanians as auxilaries in their armies. Gemindinas of Lithuania, himself of disputed Rurikid origins, married Jewna of Polotsk, and by intermarriage and conquest of Halych-Volhynia, he assumed the title "Gedeminne Dei gratia Letwinorum et multorum Ruthenorum rex" Gediminas, by the grace of God King of the Lithuanians and many Russians/Ruthenians," annexing Polotsk in 1307, and in the 1320's annexing Halych-Volhynia by his marriage ties to its dynasty (including his daughter's marriage to its last king Andrew), and installing his brother Fiodor at Kiev, and adopting the local variant of East Slavic as its official language, which it served as until 1696.
In short, Kievan Rus was a loose federation with a nominal head, which, when that head was removed, left several regional successors who claimed to represent the whole patrimony. As such, I say it never split in two, because the parts, Moscow and Galicia, claimed to be the whole. On that basis Moscow took over Novgorod, and Lithuania inherited Galicia and Polotsk's claims. While Lithuania ruled Kiev, and Poland Galicia, Kiev and Galicia claimed to be Rus'/Ruthenian and were spoken as such. Moscow's dynasty, the sole survivors of the Rurikid House of Kiev, claimed that Kiev and Galicia were occupied territory. Only with the ascension in 1633 of Met. Peter Movila/Mohyla as Metropolitan of Kiev, Halych and All Rus', a Romanian-Hungarian loyal to the Polish-Lithuanian Commonwealth but determined to restoring and perserving Orthodoxy within it do we see a dividing opposition to Russia, Met. Peter styling himself as the successor to St. Vladimir. This continues with the Hetmanate, which however throws its lot in with the Russian Tsar as successor in 1654.
2) Did the capitol move to Moscow before or after the split? If before, why?
Like Old and New Rome, Kiev retained its prestige after its actual importance had disappeared. So its primate had to live in Moscow when Kiev was unsafe, but he kept the old title.
3) When did Moscow receive its own Patriarch and why?
In 1589, because Third Rome was the only Orthodox power standing.
4) Was this before or after the capitol was moved?
After all the rest had fallen, and Moscow was left standing as sole remaining capital.
5) What needs to happen for Kiev to have its own legitimate primate today?
Things be settled between Moscow, and between Moscow and Constantinople.