In Latin class, we are translating parts of Caesar's Commentaries on the Gallic War. One thing that struck me as odd is how Caesar describes the customs of the Gauls, especially in Book 6, where he briefly describes the Druids. He mentions in chapter 14 of Book 6 that the Druids use Greek characters in writing. Greece is rather far from Gaul, so how did the Gauls and their Druids come into contact with the Greek language and learn it by the time of Caesar's campaign?
They most probably got that knowledge from cultural exchange with the greek city-states from southern France, like Massilia (Marseille), which was founded around 600 BCE and had plenty of relations with the sorrounding celtic tribes: https://en.wikipedia.org/wiki/History_of_Marseille
Its quite possible, but don't be so credulous of Caesar's judgement and reporting. Not everything he's written has turned out to be 100% accurate.
I don't know about the nuances of the original Latin, but that translation reads "they use Greek characters". That's not the same thing as using the Greek language. Right now, this post is using Latin characters. Its written in English, but it is using a character set called Latin-1.
So its quite possible what this is saying is that he found them doing some writing in an alphabet borrowed from the Greek alphabet. Latin itself borrowed its alphabet indirectly from Greek (as did arguably most alphabets in use today), so this isn't exactly remarkable. It also doesn't mean they borrowed those glyphs directly from the Greeks.
However, if the Gauls wanted to borrow the Greek alphabet, they certainly had the opportunity to do so. Only 200 years prior they controlled territory running all the way from the Atlantic nearly to the Black Sea, with enclaves on the Black Sea coast, and in Anatolia, and were invading Greece proper. 100 years before Ceasar was born, a Roman army was fighting Gauls in Anatolia (Galatia). Those Gauls at least were surrounded by Greek speakers. There were reportedly still Gallic tribes living around the Danube by Caesar's time.
Celtic domains at their maximum extent, around 270BC. info/source/key
What we know historically after Caesar is that the oldest surviving writing we have from the Gauls comes from more that 100 years after his time, and nearly everyone seems to agree that it was Latin-derived.
Before Caesar, the only written Celtic language I could find known exemplars for was Celtiberian (as the name implies, spoken in modern Spain). Its script appears to have been borrowed from the Iberian script*, which was probably derived from Phoenician (making it sort of sibling script of Greek). So this is another possible vector for the script Caesar observed. Its not Greek, but it might have looked it to him.
* - The Iberian language itself was not Indo-European. Its possibly related to Basque
Other answers indicate the prevalence of colonization, I add this one to suggest a reason why remote colonies would be founded (it's a long way from Britain and France to Greece).
If you want to make bronze you must have tin (and copper).
Europe has very few sources of tin. Therefore, throughout ancient times it was imported long distances from the known tin mining districts of antiquity.
Cornwall, Devon and Brittany and were one of few major sources of tin in Europe. And by the time Caesar came to Gaul these sites had been mined for thousands of years.
By studying the isotopes of ancient bronze objects it has been demonstrated that the tin often has origins very very far from the point that the object has been discovered, even when the design of the object matches the local culture, suggesting the object was local and the tin was imported.
First of all, there is inscriptional evidence that the Eastern Greek Alphabet was indeed used to write the Gaulish language in gallia narbonensis. The actual alphabet and some inscriptions are shown in the Wikipedia article on the Gaulish language. The reason for the adoption of this alphabet is obviously the Greek colony at Marseille.
This was not the only alphabet used for the Gaulish language, in gallia cisalpina a different alphabet (Lepontic) was used. Later, the Gauls adopted the Latin alphabet with the addition of a special letter named tau gallicum.
In addition to the answer already given, the Celts were certainly aware of the Greeks by the 3rd century BCE, as they had formed a colony in Asia minor and were slowly being Hellenized. It's not beyond the realms of possibility that the Celts in Anatolia kept in contact, due to trade or personal relationships, with those still in Gaul, and in the process exchanging various aspects of Greek culture.