The challenge, with a POD after the Mongol conquest of Kievan Rus, have a state form that:
-calls itself "Russia" and sees itself as a successor to Kievan Rus
-has its capital in Kiev or another Ukrainian city
-has Ukrainian (or something recognizably close to Ukrainian) as its official, standard language
-eventually reaches a size and shape roughly comparable to the OTL Russian Empire
Go!
As it has been pointed out above in this thread it would have been extremely difficult to do even impossible without the help of extraterrestrial flying mammals.
But on the other hand there is such thing as the Great Man. The Mongols before Chengiz Khan were a miserable bunch of nomads badly suffering from most of their neighbors. But after Chengiz Khan they became world conquerors.
So if we give the Ukrainians the Great Man of similar charisma that might help.
The best shot the Ukrainians made was under Bohdan Khmelnytsky - the so called Khmelnytsky Uprising
http://en.wikipedia.org/wiki/Khmelnitsky_Uprising
He was good. I'll give him that. But not good enough.
So let's make him as talented as Napoleon, Caesar, Chengiz Khan all put together.
So he wins all(!) his OTL battles against Poland. He devastates Poland killing, burning right and left together with his ally Crimean Tartars. And at some moment he realizes that Poland is so depopulated that this country will not be a threat for generations to come and he starts to look thoughtfully at the Crimean Tartar Army which is happily plundering nearby.
And one day he treacherously attacks it after giving some gifts or something and massacres these natural enemy of the Ukranian people killing the Khan and all the prominent beys.
The remnants of butchered Tartars are fleeing to steppe from Poland through Ukraine harassed from all sides by the Cossacks and peasants.
Bohdan Khmelnytsky makes a treaty with what left from Poland. This treaty is humiliating for the Polish honor but they are happy that Ukraine is distracted by the Crimean Tartars and so able to keep its souverenity.
And there goes a long Crimean War where Bohdan Khmelnytsky annihilated the nomad threat which hung over Rus for so many years. He gloriously defeated the forces of Ottoman Turks and so gained a worldwide reputation as a general.
Most of the Tartars are murdered, some sold to slave markets of Porta. But some Tartars smelled the winner and switched sides before the end of the war. These Tartars as Lipka Tartars of Poland became some kind of warrior caste inside the Ucranian state they were settled all over Ukraine and since then played an important role in the Ukrainian army as light cavalry which was so needed. And they were loyal personally to Khmelnytsky and his dynasty and somehow couterbalansed the Cossacks.
By the end of the Crimean War Bohdan Khmelnytsky was no longer a Hetman. He became a ruler of Rus, a champion of Orthodoxy, a hegemon in Slavic world.
I am not sure if he would call himself tsar, Caesar at this time. The Ukrainians are closer to Europe and they would not want confrontation with HRE because of this title. I think that Bohdan Khmelnytsky would style himself after the kings of Poland, his role model in titulature and court manners.
And now comes the most interesting part - most naturally Bohdan Khmelnytsky feels himself as a leader of all Rus. Kiev is an ancient capital of Kievan Rus and all that stuff. And that does not meet an understanding in Moscow for obvious reasons.
The clash is inevitable and choosing the right moment Bohdan Khmelnytsky invades Moscow Russia with the biggest army this part of the world ever seen. Of course the Muscovites give him a few battles and some cities resist the invaders. Of course Khmelnytsky wins these battles. But contrary to his reputation of ruthless conqueror in Orthodox Russia he shows himself as a most merciful Christian - he personally help the wounded Russians and publicly cry that he is so sorry that Orthodox Christians have to kill each other in this war.
That strikes the cord and Moscow surrenders without a fight. Even a tsar with his family goes to Bohdan Khmelnytsky and asks him for mercy which is granted.
The Sobor of all Russia is gathered and the former tsar publicly abdicates and ask all the land to make Bohdan Khmelnytsky the tsar of all Russia. There are a lot of miracles which prove that Bohdan Khmelnytsky as a champion of Orthodoxy who defeated innumerable hosts of the Polish Catholics and Muslim Tartars is the true tsar destined to rule over all of Rus.
Since then Kiev is the capital of all Rus and the place of Orthodox Patriarch. Some of the Moscow boyars are ordered to live there taking all their riches with them.
The rest of his life Bohdan Khmelnytsky devotes to consolidate his power over his vast Empire. There were a few riots in Russia after one of which all the family of the previous tsar was secretly executed by the angry Khmelnytsky. The defeated Russian rebels were moved in mass to settle the newly conquered Crimean lands and the Crimean Tatars were given lands in Russia.
But it was not the time of nationalism. The Moscovites did not use the word "Russians" for self identification, they used the world (Orthodox) "Christians".
So "us and them" was not "Russians against Ukrainians" it was "The Orthodox against Non-Orthodox".
I am not sure though that this country would be called "Russia" though. Probably it would be close to "Rus" or something like that.
) plus parts of Russia proper. Possibly that's when these 3 languages probably started to split. Ukrainian most likely grew as well "thanks" to civilizational decline after Mongol conquests (eastern Ukraine was scarcely populated well until XVIII/XIX century). Last thing that added to differences was settlement of Ukraine by Polish magnates. 
