There's three different answers to your question, depending on your definition of "Celt".

In a modern definition of the word, Celtic generally means a speaker of an Indo-European language part of a group including some being still spoken such as Welsh or Irish, some that disappeared such as Celtiberic or Gaulish. How do we know Gaulish was a Celtic language? Essentially from ancient sources : Greek and Romans authors mentions the name of rulers, peoples, towns, etc. they met. These aren't the same ancient source at disposal, tough, as contrary to a common belief, Gauls did wrote their language down and only refrained themselves to use it for preserving "high cultural" productions (such as epics, songs, theology, etc.). Various scripts (Gallo-Etruscan in northern Italy; Gallo-Greek in southern Gaul; Gallo-Latin in Gaul) were used to write down dedications, mark of ownership, advertising, coins, funeral stones, calendars, curses, etc.

A sentence such as CΕΓΟΜΑΡΟC ΟΥΙΛΛΟΝΕΟC ΤΟΟΥΤΙΟΥC ΝΑΜΑΥCΑΤΙC ΕΙѠΠΟΥ·ΒΗΛΗ CΑΜΙ·CΟCΙΝ ΝΕΜΗΤΟΝ^(1); while nonsensical in Greek, have words that make sense when compared to, for instance, Irish : toouta, from which tooutios is derivated, is close from Old Irish túath; and while both terms might have different meanings in their own historical context, they both name a human group we translate as tribe. Trough hundreds of exemples, we can safely say Gaulish is a Celtic language and giving the various sources at disposal, we can safely say that Gaul was entirely speaking Gaulish.

Well, not entirely... One small region of indomitable Aquitains still held out against its neighbours. And contacts weren't easy for the Celtic peoples of Volcae, Netobriges, Petrocorii and Bituriges...The region (very) roughly delimited by the Garonne river was inhabited by peoples speaking a different set of languages, probably related to modern Basque. We don't know much about Aquitains, apart than they were already considered of small importance and closer to peoples across western Pyrenees in Antiquity.

Another, smaller, group of non-Celtic speakers could be found in southern Languedoc (roughly modern Roussillon and coastal Aude) : since the VIth century, they were progressively Iberized along the coastal tradeway, speaking a non-Indo European language called Iberian attested trough Iberian scripts evidence in the region; and described as Iberians by ancient Greek scholars. But apart these fairly peripheral areas, it's generally assumed that Gauls spoke a Celtic language we call Gaulish.

Ancient Celts are generally held to be carriers of several archeological cultures across pre-Roman Europe, most famously Halstatt and La Tène^(2) : it becomes harder to assert beyond the Iron Age, but it's possible that both-or-either Urnfield Culture and Atlantic Bronze Age were regions of development of proto-Celtic languages. Overall, Gauls seems to be essentially part of these successive cultural horizons especially as inhabited along important trade roads (polarized along the Rhône, Saône, Loire, Seine and Rhine rivers connecting the Mediterranean basin with Northern Europe). As such they're counted among the ancient Celts, a group of protohistorical peoples sharing beyond their languages (which aren't always well attested for, especially beyond the Rhine and the Alps) a set of archeological features (art, crafting techniques, etc.).

Associating both linguistic and archeological identifications, Gauls are considered being a Celtic people, one among many. In short, all Gauls were Celts while not all Celts were Gauls. Making the difference between Gauls and other Celtic people can be relatively fair (Celtiberian language is quite distinct from Gaulish; and while British language seems to have been close to Gaulish, its speakers had their own cultural particularities) but it can be tricky as well : Germania in particular seems to have been inhabited by Celtic peoples sharing a lot of cultural and linguistic features (Ancient Germans not being necessarily Germanic-speakers), to the point it's suspected that Caesar created in his Commentaries a radical cultural difference between Gauls and Germans where it didn't existed, at least not as importantly; something that the Roman conquest, its consequences on local peoples, and the establishment of a clearly delimited border reinforced.

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^(1) Segomaros, son of Uilleos, citizen of Nimes, gives to Belisama this sacred groove.

^(2) It doesn't mean at the latest that people that would have adopted these material features would have been necessarily entirely or party Celtic-speaking (especially in northern and central Germany before the turn of the Ist millenium), or that peoples from different archaeologic cultures wouldn't be Celtic-speaking (notably Old Ireland). It's more of a broad association between material and immaterial culture of protohistoric peoples and thus more fuzzy further you go from ancient sources.