This is a good question and it is not easy to answer. As you say, the Greeks thought of each other as members of the same group. They spoke the same language and honoured the same gods. But they lived in hundreds of small states, and each state made its own laws and wars. There was no government of all Greeks. The cities and federations of the Greek world chose their own friends and enemies.

Since most Greek states were small and not very rich, most of their problems were not with non-Greek peoples far away, but with their own Greek neighbours. Raiding, piracy and disputes over lands on the border between states could often cause wars. These raids could be terrible; the attackers would not just steal cattle and sheep but also destroy farms and enslave women and children. We can imagine that they could make Greeks hate each other. They also fought over insults and offences against the gods. I wrote more about the causes of Greek wars here.

How often did they fight? Some historians like to say that they were fighting all the time. They quote Plato, who wrote that war was actually the normal state of things, and peace was only a short pause. They say that several peace treaties between states were for a set time. For example, in 446 BC, Athens and Sparta made a peace for 30 years. In 421 BC, Athens and Thebes made a peace which had to be renewed every 10 days. If peace was only supposed to last for such a short time, then maybe war was normal; maybe there was always war between all Greeks all the time, unless they made a temporary peace.

But other historians disagree. They say the Greeks knew that war was awful, and always tried to avoid it. They quote Xenophon, who wrote that if states must fight wars, they should do so only briefly, and go back to peace as quickly as they can. They say peace treaties weren't short because war was normal, but because these treaties also included alliances, and most states did not want to be stuck in an alliance forever. That is why they made the treaty short. The peace might last much longer after that.

Most Greek states probably did not fight very often. But the ones that we read about were the ones that spent a lot of time attacking others and fighting each other. States that became powerful would often try to get rich through war, to gain status, and to rule over other states. The Greeks in those other states would join together to stop them. This is what happened when Sparta fought Athens (431-404 BC), and again when Athens, Thebes, Corinth and Argos made an alliance to fight Sparta (395-386 BC).

Some Greeks hated each other very much. Athens and Aegina fought each other for more than a century; so did Samos and Miletos. Herodotos says the Thessalians and the Phokians hated each other so much that, if one of them chose to resist the Persian invasion, the other would immediately choose the Persian side. But states could also be friends for just as long, like Athens and Plataia. Often their choice of friends and enemies was based on what they needed at the moment. Athens fought Sparta in several wars, but in 370 BC they made an alliance to fight the Thebans together, because Thebes had become a threat to them both.

After the Persian Wars, some Greeks were eager to stop the fighting between Greeks and to get all the states to join forces against Persia. They said it was wrong for Greeks to kill and enslave other Greeks, when there was an enemy in the East who had attacked them first. But these so-called panhellenists (the word means "all-Greeks") never got enough power to make their war against Persia happen. The Greek states were much too busy looking after themselves and their own wealth. They would not join forces against Persia until the Macedonians forced them to. And even then, there were Greeks rebelling against other Greeks and Macedonians while Alexander was in the East!

In short, even though the Ancient Greeks thought all Greeks were similar, they were never united in a single state, and always kept fighting each other until the Romans conquered them.