Why didn't the silk road run north of the Black Sea?

There was actually a trade route through Ukraine - "из варяг в греки" - that is "From Varangians to the Greeks". You could travel from Byzantine Empire to Scandinavia through Kiev and back. But traveling further East would be problematic: You'll face first the warlike Turkic tribes (Kipchaks and Pechenegs) then the warlike Circassians and Chechens and then a big empty space with no city for thousands of kilometers.

In those days, most of the people of the "Ukraine" lived in the western part of the province (around Kiev). The eastern part, which abuts on Kazakhtstan, was mostly "deserted," except for the fierce nomads that later became the "Cossacks."

The easiest route from Persia (the main "terminus") to Kiev was via northern Turkey, and from there by water across the Black Sea and up the Dnepr. Although that was an arduous route, it was still safer BY FAR than via Kazakhstan and the eastern Ukraine. Only the nomads could successfully "navigate" the deserts and steppes of the latter territories. And they didn't have the trading skills to manage a piece of the silk trade.

My family came from Western Ukraine and my grandfather attended University in Chernotsy and Vienna. He said that historically our family were merchants on a branch of the Silk Road that ran through Ukraine. Ukraine was a bulwark in the hundreds of years of warfare between Europe and Turkey and it is absurd to think that a substantial amount of trade did not move preferentially through Ukraine rather than Turkish controlled territory. There are many possible explanations for the lack of recognition of all things related to Ukrainian history. The most likely is that Russian state "scholars" in the time of the Russian and Soviet empires set about obliterating Ukrainian history to make it appear as if Ukraine was a minor subset of Russia and Western scholars have simply accepted much of the propaganda. Another is simply the lack of interest or even open hostility to Eastern European history. Inane comments like there was nothing there in Ukraine, are typical. Why don't most Westerners know about the centuries long war between Ukraine and Turkey, or why is it assumed that mongols dominated Ukraine for centuries but left very little genetic trace while countries like Iraq and Iran, which carry the indelible mark of Mongol genes, are poetically referred to as Persian. Go figure.

The Silk Way was not a single road, but rather a net of roads. And the ways Amudarja/Uzboj (Amudarja went to Caspian Sea till 16 Century, for example) - Caspian Sea - Volga - Don - Azov sea - Black sea - Konstantinople (variant: Aral-Caspian Sea by foot) was in use -especially for long periods when Amudarja was switched to the Caspian Sea and some stable state as Khazars sat on the lower Volga. Because it was a quick way - the only problem was to carry boats of goods from Volga to Don).

The southern way, anyway, was more popular, having worse roads (through mountains), but more cities on the way. But it is hard to say what was the reasson - cities, or the Silk way. Because after Vasco da Gama The Silk way died and these cities degenerated, too.

So, some parts of the Silk Way did pass along the North of the Black Sea. The Genoa town of Cafa (Feodosia now) lived mostly from this very part of the Silk Way. Marco Polo finished his way there.

But to move goods from East to West in the stripe to the North of the Black Sea was always a bad idea. Because it was very slow (carts and horses are slower than ships) and very complicated and expensive - you would have to cross many large rivers and to pay to the owners of ferries or/and places around them.

Just take a look at any political map, let it be Classical period, or early Medieval times. When travelling to China you need water, supplies of food, fodder, etc. Also it's safer to spend a night in a city or some kind of inn instead of open steppe spaces. Then what Joe mentioned, between the cities you've got roads, which again - are safer.
South of Black Sea you'd travel through Anatolia, Iran, Sogdiana, Bukhara, Samarkand, etc. pretty civilised and relatively safe areas.
And north of Black Sea? What is now Ukraine and Kazakhstan at the time was just a one BIG empty open steppe space, hard to find any road, any food, any water (not to mention the nomads which may be hostile).