If by involved we mean part of the actual fighting in the Pacific theater, the answer is not very. There were some Free French ships in the Pacific, but they mostly ran escort and transport duty. They had no battleships in the Pacific until very near the end of the war and no carriersof which to speak (well sort of Béarn, but they were under house arrest in Martinique).

If by involved we mean actively participating in the Allied war effort, then we can say very involved. The French colonies in the Pacific were some of the first to answer De Gaulle's call for Free France and volunteers and troops were on their way out of the Pacific by the end of 1940. The main thrust of France's involvement was material and strategic aid in the Pacific. The colonial holdings of New Caledonia and the New Hebrides (which they shared with the British) were just outside of the Japanese advance and were well positioned to be front line bases for the allies. Indeed New Caledonia became the site of a massive allied airfield during the war and Nouméa, the colonial capital, became the headquarters for the US Navy/Army command in the South Pacific. Before the war their had been some fear as well that Japan would move to quickly annex New Caledonia because of its great mineral wealth, especially its nickel mining industry. In the 1920's there had even been rumors that Japanese mining workers were going to stage a revolt and annex the island, or that Japanese companies would take over the mining industry and dominate the island's economy (neither of these were true of course). And further afield from the front, Bora Bora in French Polynesia was an important airfield and resupply base for the central Pacific.