The first example that I can find of the phrase is in the French author Eugène Sue's novel Memoirs of Matilda, which was translated into English by D. G. Osbourne and published in 1846:
And then revenge is very good eaten cold, as the vulgar say.
The italics are from the text, which implies that the phrase was already in use when the novel was written. As always with translations, it is a moot point as to who can claim authorship of the proverb as an English phrase - the translator, who was the first to use the expression in English, or the original author.
Wherever it can be said to have originated, the proverb struck a chord in the English-speaking world. More recently, it has been called into use in three screen classics:
Kind Hearts and Coronets, 1949: "Revenge is a dish which people of taste prefer to eat cold."
The Godfather, 1969: Don Corleone nodded. "Revenge is a dish that tastes best when it is cold," he said.
Star Trek II, The Wrath of Kahn, 1982: "Kirk, old friend, do you know the Klingon proverb, 'Revenge is a dish best served cold'?"