Alexander was one of the historical figures who truly deserves the honorific “the Great.” He was a military genius and a courageous warrior, personally leading his armies in battle and fighting on despite being wounded on several occasions. He was a charismatic and inspirational leader who won the loyalty not only of his Macedonian countrymen, but the Greeks and, most remarkably, the people of the Persian Empire whom he conquered. He was also driven by incredible ambition; tutored by none other than Aristotle in his youth, he modeled himself on the legendary Greek hero Achilles, hoping to not only match but to surpass Achilles' prowess in battle. He became a legend in his own life - he was worshiped as a god by many of his subjects and even his Greek subjects came to venerate him as one of the greatest leaders of all time.
Alexander's conquests began almost immediately after seizing the throne. He first ruthlessly killed off his rivals and enemies in Macedon and Greece, executing nobles he suspected of treason, and then leading an army back through Macedon, crushing the Thracian tribes of the north who threatened to defect. Some of the Greek poleis rose up, hoping to end Macedonian control almost as soon as it had begun, but Alexander returned to reconquer the rebellious Greek cities. In the case of the city of Thebes, for instance, Alexander let the Thebans know that, by rebelling, they had signed their own death warrant and he refused to accept their surrender, sacking the city and slaughtering thousands of its inhabitants as a warning to the rest of Greece.
By 334 BCE, two years after he became king, Alexander was thoroughly in control of Greece. He immediately embarked on his father's mission to attack Persia, leading a relatively small army (of about 45,000 men) into Persian territory - note how much smaller this army was than the Persian one had been a century earlier, when Xerxes had invaded with over 200,000 soldiers. He immediately engaged Persian forces and started winning battles, securing Anatolia and the rich Greek port cities in 333 BCE and Syria in 332 BCE. In almost every major battle, Alexander personally led the cavalry, a quality that inspired loyalty and confidence in his men.
His success against the Persians can be explained in part by the fact that the Persian technique of calling up their armies was too slow. Even though Alexander had arrived in Anatolia with only 45,000 men, against a potential Persian army of close to 300,000, far fewer troops were actually available to the Persians at any one time during the first years of Alexander’s campaign. Instead, the first two years of the invasion consisted of Macedonian and Greek forces engaging with smaller Persian armies, some of which even included Greek mercenaries. Alexander’s forces succeeded in conquering Persian territory piecemeal, taking key fortresses and cities, seizing supplies, and fighting off Persian counter-attacks; even with its overall military superiority, the Persian Empire could not focus its full might against the Greeks until much of the western empire had already been lost. In addition, Alexander was happy to offer alliances and concessions to Persian subjects who surrendered, sometimes even honoring with lands and positions those who had fought against him and lost honorably. In sum, conquest by Alexander was not experienced as a disaster for most Persian subjects, merely a shift in rulership.
In 332 BCE, the Persian king, Darius III, tried to make peace with Alexander and (supposedly - there is reason to believe that this episode was invented by Greek propagandists afterwards) offered him his daughter in marriage, along with the entire western half of the Persian Empire. Alexander refused and marched into Egypt, where he was welcomed as a divine figure and liberator from Persia. Alexander made a point of visiting the key Egyptian temples and paying his respects to the Egyptian gods (he identified the chief Egyptian deity Amun-Ra with Zeus, father of the Greek gods), which certainly eased his acceptance by the Egyptians. In the meantime, Darius III succeeded in raising the entire strength of the Persian army, knowing that a final showdown was inevitable.
From Egypt, the Greek armies headed east, defeating the Persians at two more major battles, culminating in 330 BCE when they seized Persepolis, the Persian capital city. There, the Greek armies looted the entire palace complex before burning it to the ground; historians have concluded that Alexander ordered the burning to force the remaining Persians who were recalcitrant to his conquest to acknowledge its finality. The wealth of Persepolis and the surrounding Persian cities paid for the entire Greek army for years to come and inspired a renaissance of building back in Greece and Macedonia, paid for with Persian gold. Darius III fled to the east but was murdered by Persian nobles, who hoped to hold on to their own independence (this did not work - Alexander painstakingly hunted down the assassins over the next few years).