Cambyses had trained for the throne by governing Babylonia for eight years by the time his father died in battle against the Massegetae. During his father's rule, he had adopted his father's dream of conquering Egypt. This he did, in the fifth year of his reign, after he had spent those years in planning -- figuring out how to get water and provisions for the troops while crossing the long expanses of desert.
His solution? He made treaties with the Arab tribes along the route. Cambyses succeeded, and became pharaoh of upper and lower Egypt, the first of the Persian monarchs ruling the African country.
See: "The Legitimacy of Cambyses and Darius as Kings of Egypt," by K. M. T. Atkinson; Journal of the American Oriental Society, Vol. 76, No. 3 (Jul. - Sep., 1956), pp. 167-177.
Whether or not Cambyses would have been able to defeat the Egyptians had their experienced monarch, Amasis, remained in place may be debated, but luckily for Cambyses, that pharaoh had died. In his place was an inexperienced pharaoh, Psammenitos (Psamtik III).
Here are excerpts from what the Greek historian Herodotus has to say in Book III of his histories (from an English public domain translation by G. C. Macaulay) about Cambyses' conquest and treatment of his enemy:
After conquering Egypt, Cambyses turned to other locations he wished to conquer, like Ethiopia and Carthage, but he failed."10. Meanwhile Psammenitos the son of Amasis was encamped at the Pelusian mouth of the Nile waiting for the coming of Cambyses....
"11. Now when the Persians had marched quite through the waterless region and were encamped near the Egyptians with design to engage battle, then the foreign mercenaries of the Egyptian king, who were Hellenes and Carians, having a quarrel with Phanes because he had brought against Egypt an army of foreign speech, contrived against him as follows:-- Phanes had children whom he had left behind in Egypt: these they brought to their camp and into the sight of their father, and they set up a mixing-bowl between the two camps, and after that they brought up the children one by one and cut their throats so that the blood ran into the bowl. Then when they had gone through the whole number of the children, they brought and poured into the bowl both wine and water, and not until the mercenaries had all drunk of the blood, did they engage battle. Then after a battle had been fought with great stubbornness, and very many had fallen of both the armies, the Egyptians at length turned to flight."Egyptians Surrender, Send Tribute, and Destroy the Mytilene Ship
"...13. The Egyptians when they turned to flight from the battle fled in disorder: and they being shut up in Memphis, Cambyses sent a ship of Mytilene up the river bearing a Persian herald, to summon the Egyptians to make terms of surrender; but they, when they saw the ship had entered into Memphis, pouring forth in a body from the fortress [13] both destroyed the ship and also tore the men in it limb from limb, and so bore them into the fortress. After this the Egyptians being besieged, in course of time surrendered themselves; and the Libyans who dwell on the borders of Egypt, being struck with terror by that which had happened to Egypt, delivered themselves up without resistance, and they both laid on themselves a tribute and sent presents: likewise also those of Kyrene and Barca, being struck with terror equally with [14] the Libyans, acted in a similar manner: and Cambyses accepted graciously the gifts which came from the Libyans, but as for those which came from the men of Kyrene, finding fault with them, as I suppose, because they were too small in amount (for the Kyrenians sent in fact five hundred pounds' weight [15] of silver), he took the silver by handfuls and scattered it with his own hand among his soldiers."Cambyses Dishonors the Pharaoh and His Family
"14. On the tenth day after that on which he received the surrender of the fortress of Memphis, Cambyses set the king of the Egyptians Psammenitos, who had been king for six months, to sit in the suburb of the city, to do him dishonour.... [H]aving arrayed his daughter in the clothing of a slave, he sent her forth with a pitcher to fetch water, and with her he sent also other maidens chosen from the daughters of the chief men, arrayed as was the daughter of the king.... next Cambyses sent his [Psammenitos'] son with two thousand Egyptians besides who were of the same age, with ropes bound round their necks and bits placed in their mouths; and these were being led away to execution to avenge the death of the Mytilenians who had been destroyed at Memphis with their ship....[T]he Royal Judges [17] had decided that for each man ten of the noblest Egyptians should lose their lives in retaliation. He then, when he saw them passing out by him and perceived that his son was leading the way [18] to die...."Cambyses Pities the Pharaoh
[T]here entered some pity into Cambyses himself, and forthwith he bade them save the life of the son of Psammenitos from among those who were being put to death, and also he bade them raise Psammenitos himself from his place in the suburb of the city and bring him into his own presence. 15. As for the son, those who went for him found that he was no longer alive, but had been cut down first of all, but Psammenitos himself they raised from his place and brought him into the presence of Cambyses, with whom he continued to live for the rest of his time without suffering any violence; and if he had known how to keep himself from meddling with mischief, he would have received Egypt so as to be ruler of it...."Cambyses Punishes the Pharaoh in Disregard of Religion of Both Countries
"As it was, however, Psammenitos devised evil and received the due reward: for he was found to be inciting the Egyptians to revolt; and when this became known to Cambyses, Psammenitos drank bull's blood and died forthwith. Thus he came to his end. 16. From Memphis Cambyses came to the city of Sais with the purpose of doing that which in fact he did: for when he had entered into the palace of Amasis, he forthwith gave command to bring the corpse of Amasis forth out of his burial-place; and when this had been accomplished, he gave command to scourge it and pluck out the hair and stab it, and to do to it dishonour in every possible way besides: and when they had done this too until they were wearied out, for the corpse being embalmed held out against the violence and did not fall to pieces in any part, Cambyses gave command to consume it with fire, enjoining thereby a thing which was not permitted by religion: for the Persians hold fire to be a god. To consume corpses with fire then is by no means according to the custom of either people, of the Persians for the reason which has been mentioned, since they say that it is not right to give the dead body of a man to a god; while the Egyptians have the belief established that fire is a living wild beast, and that it devours everything which it catches, and when it is satiated with the food it dies itself together with that which it devours: but it is by no means their custom to give the corpse of a man to wild beasts, for which reason they embalm it, that it may not be eaten by worms as it lies in the tomb. Thus then Cambyses was enjoining them to do that which is not permitted by the customs of either people."
