One of the most important corner stones of British is the protection of her trade. The main way of doing so was to ensure that there was not any sort of political hegemony that would threaten Britain. This would be reflected throughout the 18th century, such as Britain's attempt to stop the French during the Wars of the Spanish Succession (see the Campaigns of Marlborough), and by siding with the Prussians during the War of Austrian Succession and the Seven Years War (both times fighting the French to stop French political hegemony). With that, the British were looking to stop any single power from gaining control of Europe. This would continue into the French Revolution.

While the British were outraged by the murder of the French aristocracy (as well as housing many of the emigrant nobility), the British experienced a change of politics when William Pitt was replaced by Henry Addington as Prime Minster, whom was more interested in creating peace. After a result of a problem with the Treaty of Aimes (which created peace with The French Consulate), William Pitt would return as Prime Minster and resume a hostile attitude with France.

Most importantly, as /u/britainopplsnef has said, Britain was the puppet master of the Wars against France. While each monarch had it's reason to fight Napoleonic France, British gold financed the wars and helped give them courage to fight the Emperor. Napoleon even was yelling at the Austrian soldiers after the Capitulation of Ulm about the dangers of British gold and the "Perfidious Albion" that got them into pointless wars.

After 1803, the British and French were in a state of constant war. It was dangerous for Britain to allow the French a hegemony over continental Europe. The British needed trade and Napoleon actively worked against it.

Britain didn't declare war on the French, the French declared war on Britain in February 1793, although Britain was preparing for war anyway and had complained to, and threatened, France about her territorial acquisitions in the Low Countries, specifically being unhappy about the opening of the Scheldt. France then pursued war against Britain, regarding Britain as a main enemy, such that the Convention declared that British troops were not to be taken prisoner. Britain remained at war with France until the Peace of Amiens, by which time General Buonaparte was head of state of France, as First Consul. So, essentially, Britain didn't go to war with Buonaparte; Britain's conflict with Buonaparte was simply part of an ongoing war with France. Neither side was faithful to the Peace of Amiens, (interesting Gilray cartoon on the Treaty here http://upload.wikimedia.org/wikipedia/commons/8/80/Gillray_-_The_First_Kiss.jpg ) with both sides showing bad faith; Britain didn't evacuate Malta or Egypt, as the Treaty demanded, and Buonaparte sent military expeditions to the West Indies and Louisiana, which Britain viewed a threat to the Atlantic trade routes, which, of course, it was. The Treaty soon broke down and in May Britain effectively resumed hostilities, without declaration of war, by seizing French and French allied shipping in British waters, declaring war the next day. Buonaparte responded by arresting all British subjects in French territory, an unheard of event. The war between Britain and France continued up to Buonaparte's abdication, Britain using proxies to fight in continental Europe, by financially and politically supporting the continental powers against Buonaparte's continual territorial expansion, whilst controlling the seas. The control of the sea led ultimately to Buonaparte's defeat, as the economic strangulation of Europe led to his attempts to enforce his "Continental System", which was seriously damaging the economies of the European powers, which in turn led to his invasions of Portugal, Spain and Russia, leading to his downfall.