Although he is usually described as Greek, this site, suggests that Clitarchus might have been Egyptian. It also mentions his father, the historian Dinon of Colophon, who wrote a History of Persia that elaborated and extended Ctesias' Persica. This is now lost, but surviving fragments show that Dinon began with Semiramis' Babylon and that he had a taste for the fabulous, so he probably did include material on the Hanging Gardens.
Clitarchus' work only survives in fragments, too, but it is thought that he completed his History of Alexander in 310 or 301 BC, using eye-witness accounts and his father's lost history—especially for the descriptions of Babylon (fragment 10). Although Pliny the Elder (Natural History 3.57-58) says that Clitarchus was in Babylon at the same time as Alexander, we have seen that this is unlikely.
Some 20 or 30 fragments of Clitarchus' works are said to have been preserved or paraphrased by Strabo, and he is also said to have been a major source for the later sections of Diodorus (i.e. not the part about the gardens) and especially for Quintus Curtius Rufus. Quintus Curtius' description of the Hanging Gardens (pasted in brown below) is usually attributed entirely to Clitarchus.
Quintus Curtius, a Roman military commander (but not noble), wrote his History of Alexander the Great of Macedonia, in 31 and 41 AD. Originally in ten books, the first two are now missing. It usually said to have been based heavily on Clitarchus (above), who completed his own History of Alexander 310-301 BC, using eye-witness accounts.
The Livius site extracts Quintus Curtius' account of Babylon and the Hanging Gardens (pasted in brown below). It comes from Book 5, chapter 1, lines 17-33 (text is from The History of Alexander, Penguin Classics, 1984, translated by John Yardley). The gardens appear in the last paragraph.
Surrounded by an armed guard, the king [Alexander] instructed the townspeople to follow at the rear of his infantry; then he entered the city on a chariot and went into the palace. The next day he made an inspection of Darius' furniture and all his treasure, but it was the city itself, with its beauty and antiquity, that commanded the attention not only of the king, but of all the Macedonians.
And with justification. Founded by Semiramis [...], its wall is constructed of small baked bricks and is cemented together with bitumen. The wall is ten meters wide and it is said that two chariots meeting on it can safely pass each other. Its height is twenty-five meters and its towers stand three meters higher again. The circumference of the whole work is 365 stades, each stade, according to the traditional account, being completed in a single day.
The buildings of the city are not contiguous to the walls but are about thirty meter's width from them, and even the city area is not completely built up - the inhabited sector covers only 275 hectares - nor do the buildings form a continuous mass, presumably because scattering them in different locations seemed safer. The rest of the land is sown and cultivated so that, in the event of attack from outside, the besieged could be supplied with produce from the soil of the city itself.
The Euphrates passes through the city, its flow confined by great embankments. Large as these structures are, behind all of them are huge pits sunk deep in the ground to take water of the river when in spate, for when its level has exceeded the top of the embankment, the flood would sweep away city buildings if there were no drain-shafts and cisterns to siphon it off. These are constructed of baked brick, the entire work cemented with bitumen.
The two parts of the city are connected by a stone bridge over the river, and this is also reckoned among the wonders of the East. For the Euphrates carries along with it a thick layer of mud and, even after digging this out to a great depth to lay the foundations, one can hardly find a solid base for a supporting structure. Moreover, there is a continuous build-up of sand which gathers around the piles supporting the bridge, impeding the flow of water, and this constriction makes the river smash against the bridge with greater violence than if it had an unimpeded passage.
The Babylonians also have a citadel 3,7 kilometers in circumference. The foundations of its turrets are sunk ten meters into the ground and the fortifications rise 24 meters above it at the highest point. On its summit are the hanging gardens, a wonder celebrated by the fables of the Greeks. They are as high as the top of the walls and owe their charm to the shade of many trees. The columns supporting the whole edifice are built of rock, and on top of them is a flat surface of squared stones strong enough to bear the deep layer of earth placed upon it and the water used for irrigating it.