Napoleon was born Napoleone Buonaparte (Corsican: Nabulione) in the Maison Bonaparte in Ajaccio on the island of Corsica, which had been sold to France by the Republic of Genoa in 1768 after a long war of independence. He was the second son of Carlo Buonaparte and Letizia Ramolino, who had 13 children together, but only eight survived early childhood. On 21 July 1771, Napoleon was baptised in Notre-Dame-de-l'Assomption Cathedral. The family belonged to the minor nobility of Corsica and had been resident on the island since the early 16th century. Their roots lay in Tuscany, Italy. Napoleon's grandfather was the Corsican politician Giuseppe Maria Buonaparte; his father Carlo was the secretary of Pascal Paoli, a Corsican revolutionary and resistance fighter, and had fought with him for Corsican independence. After initial successes, the insurgents were crushed at the Battle of Pontenuovo and Paoli went into exile in Britain.

So yes, he was a French citizen since birth.