The specific kingdom in question is generally known as the Greco-Baktrian kingdom in history, which is because Baktria existed before Alexander and it was ruled by Greeks only for that specific period. There is a big caveat before I explain further, which is that there's a lot we don't know either about Baktria or the Greco-Baktrian kingdom, but I'll try to give you some information that would be helpful.

Baktria had a distinct Iranian-speaking culture as of the time of Alexander's conquests, different from that of the Persians. They were part of the Achaemenid Persian Empire, but we aren't really aware of their history before the Persians conquered them; was Baktria an independent kingdom, was it part of a larger one in Central Asia, was it several different states, we can't say for sure. Bactria was very arid, with its cities and agriculture reliant on oases and irrigation canals taking river water in order to grow crops. They did so extremely successfully, with Baktria being later renowned among western Greeks for its numerous cities, and wealth in general, and similar reports passed their way eastwards to the Han court. Despite this famed urbanism, the region was also a key zone of interaction between the nomadic cultures of the deeper steppe and settled people, and we think that the Achaemenids had already begun to use Baktria to house and recruit high quality cavalry from nearby areas.

Alexander found conquering this part of the Achaemenid extremely difficult, and spent around two years conquering and pacifying Bactria and the surrounding regions. His personal connection to Bactria became his wife Roxanne, and his son Alexander IV. This connection was retained by his officer Seleukos, who founded the Seleukid Empire. His wife, Apame, was also Baktrian, and allegedly Seleukos was one of the few of Alexander's generals who retained their Iranian wives after Alexander's death. When Seleukos gave up the Indus' territories he had inherited, and large portions of Arachosia, to the newly emerged Mauyran Empire in India it's interesting that Baktria was specifically retained, and it seems the Seleukids invested a lot of time and money in developing and holding onto Baktria- the Seleukids created new urban areas, rebuilt older ones, patronised major local temples, and imported Hellenic colonists into the region. Antiochus I, the son of Seleukos, governed Bactria and nearby regions during his co-regency with his father.

The loss of Baktria in the 230s-220s BCE was quite a blow to the Seleukids, and was enabled by a major civil war between rival claimants to the throne. The newly independent kingdom was founded by one Diodotos, who had been the governor at this time, but his son Diodotus II was overthrown by one Euthydemos, the new dynasty hadn't lasted very long. Antiochus III's expedition eastwards to recover lost territory had Baktria as a major objective. He allegedly besieged the capital of Baktria, known as Baktra to the Greeks and known in modern times as Balkh, for a year, and we think he also besieged and sacked at least one other major Baktrian city at this time (the archaeological evidence fits that explanation better than the earlier suggestion that Scythians sacked it). However, Antiochus III did not successfully recover Baktria, and instead recognised Euthydemos as king over Baktria (with a lesser rank and status than himself, of course).

The Greco-Baktrian kingdom shows signs of having begun a kind of cultural fusion. The one major city of theirs that's been significantly excavated, usually called Ai Khanoum (it's the modern name for the area), is very much a Greek city but not without local influences. The city had, by the end of its life in the 130s BCE, a gymnasion, a major palace complex that took up a large chunk of the city, a large theatre, and extremely imposing walls fortified with towers. However, its two major temples were more local than Hellenic, one of which continued to operate an Aramaic bureacracy for some time after Alexander's conquest. We have also discovered a temple to the deified river Oxus itself in which at least one dedication was written in Greek by someone with an Iranian name. A later indication of the influence of Greek culture on Baktria as a whole was the fact that the Baktrian language was first written down in a modified version of the Greek alphabet, and references to Greek deities in the area continued for some three centuries after the collapse of the Greco-Baktrian kingdom.

As for connections with India, there had also been Hellenic communities established in the Indus and in Arachosia, but they had been conquered by the Mauryans and had been governed as part of that Empire for some time. However, by the dawn of the 2nd century BCE the Mauryans had pretty much collapsed, and the Greco-Baktrians started launching expeditions against the Indus region. There may have been one launched around 190 BCE, they certainly launched a major one around 180 BCE, one which conquered swathes of North-western India and modern Pakistan (the exact extent is very difficult to tell and keenly debated). However, not long after this conquest the Greco-Baktrian state split into two, with the original Euthydemid dynasty overthrown by that of Eucratides, and the Greek-ruled areas of India splitting into loose coalitions of individual kingdoms (we're pretty sure anyway). The exact politics of this time are extremely difficult to penetrate at times, for example we can evidence tribute paid in Indian currency still reaching cities in Baktria in the period right before its conquest by Scythians. The most famous Greek king of this time is King Menander, who is believed to have ruled a large kingdom based on the capital of Taxila and who may have controlled some parts of Baktria until it fell to Scythians. Menander is the main character of the Buddhist text the Milinda Panha, and is believed to have started the fertile encounter between Greek culture and the Buddhist religion, as evidenced perhaps by his bilingual coins issued during his reign.

When the Greco-Baktrian kingdom fell, it fell hard. The city at Ai Khanoum that was discovered was totally sacked around 140-130 BCE and was never again inhabited as a city, only ever as a fortress on what had once been its citadel. Chinese accounts, written by authors exploring out west for the first time in this period, definitely speak of its prior rulers remaining in the cities, but are entirely clear that they were now under the control of nomadic rulers. This was the end of Greek rule in Baktria, which had lasted around 160 years, though not in India where various small kingdoms continued to exist into the 1st century CE. As I indicated before, it was far from the end of Greeks or Greek culture in the area, but they would never again rule this part of the world.

Baktria was rich, powerful, home to populations of Greeks and Baktrians along with, likely as not, Scythians and some number of Indians, expansionist, and culturally unusual. It most certainly cared about cavalry and made efforts to settle cavalrymen both in urban areas and in key grazing lands. It had some degree of reknown and mystery back in the Mediterranean. But it was also obscure, to the Mediterranean world, where it held almost no connection to goings on and where all of its major wars and deeds had no intersection with their own. It continues to be obscure, arguably far less so since the 90s but obscure nonetheless. However, it's an extremely fascinating and particular culture that I greatly enjoy studying, and care about a lot as you can probably tell.