Elizabeth had two main problems concerning France:
Elizabeth’s predecessor, Queen Mary I, had been in an anti-French alliance with her husband, King Philip of Spain. During Mary’s reign she lost the strategic town of Calais which had been ruled by England for hundreds of years, so this meant that France now controlled the entire northern coastline of France and left England more vulnerable of an attack.
Mary, Queen of Scots had been married to Francis II, King of France before he died so the French were supporters of Mary and her claim to the English throne. When Elizabeth eventually had her executed, Elizabeth feared serious repercussions from the French.
Despite initially being a threat to England, civil war in France from 1562 meant the French monarchs were too preoccupied with home affairs to pose a threat to Elizabeth. Even when Mary, Queen of Scots was executed, the French King, Henry III, kept on friendly terms with Elizabeth.
Although trying, and failing, to take back Calais, Elizabeth managed to keep peace and a good relationship with France going, she even entered negotiations to marry Duke of Alencon to help strengthen this.