1. The Greek homeland was in the southernmost part of the Balkan Peninsula, as well as the islands in the Aegean Sea. The northernmost point where the Greeks had territory was to the Mount Olympus. The location of the Greek homeland relative to the surrounding area comes to be in the central western part of Mediterranean Sea. It is still not totally clear from where did the Greeks came in this area, but it is thought that it was probably from Asia Minor, with their ancestors being known as Doric people.
*note: Macedonia was not part of the Greek city-states. The Macedonians had different ethnic background, different culture, and different language than the Greeks.
2. There's a common misconception that Alexander gained control of Greece, but that is actually not true. It was Alexander's father, Philip II of Macedon who conquered the Greek city-states, while Alexander inherited them and just strengthened the control over them as the Greeks openly hated the Macedonians who they considered as barbarians, so it was the biggest humiliation to be ruled by them. Also, it was no all Greek city-states that were conquered, with Sparta being left alone and independent.
3. Soon after he succeeded the empire, Alexander started to organize military campaigns. He started with the Balkans, and managed to strengthen his positions in Thrace, southern Illiria, as well as Greece. Than he moved toward east, having numerous battles with the Persians, and constantly pushing them toward east. By the year of 332 BCE, the Macedonian Empire managed to include the southern half of the Balkans, Asia Minor, the western part of the Middle East, and Egypt.
4. The Macedonian army and the Persian army had lot of battles until the final outcome was decided. The Macedonian army, despite being significantly smaller, managed to win battle by battle, causing lot of damage to the Persians. Darius got very mad about how things were going, so he gathered enormous army, helped even by thousands of Greek soldiers who fought alongside the Persians despite having wars with them recently, but they hated the Macedonians more. Alexander showed his tactical abilities on the battlefield, by managing to outmaneuver the Persian army and destroy it at Gaugamela, with Darius having to flee from the scene so that he doesn't end up dead. With the emperor of Persia out of the way, this battle turned out to be decisive, and Alexander victoriously marched through Persia, claiming it as part of Macedonia.
5. After the win of Gaugamela, Alexander had the Persian Empire under his feet. He marched through it with his army, claiming the Persian lands to himself. The first stop that Alexander made was in the capital of Persia, Babylon, where he officially presented himself to the Persians as their new king. He then continued east to Susa, the former capital of Persepolis, before turning northwards to Pasargadae. The final major Persian city which was visited by Alexander was the capital of Bactria, Bactra, after which he started to move toward southeast and reached the Indian subcontinent.
6. Alexander's empire managed to stretch from the western Balkan as far east as the northwestern part of India, making it the largest empire the world had seen until then, and in fact it is still one of the largest empire to have ever existed. When compared the maximum size that the empire reached with the maps prior to the final conquests, it is easily noticeable that the Macedonian Empire has increased significantly, and on top of it, in a very short period of time. The empire managed to become so large in such a short period of time because the majority of the territory conquered by Alexander was actually controlled by Persia, so once Persia was defeated, the empire gained enormous territory over night.