To deliberately say one thing and mean another, to be hypocritical.
In the longstanding tradition of many Native American tribes, “speaking with a forked tongue” meant lying, and once a person had been shown to speak with a forked tongue, he was no longer considered worthy of trust. As Tonto might have said, “White man say one thing, but mean another.”
This phrase was adopted by European Americans around the time of the Revolution, and there are many references from the early 19th century — often reporting on American officers who sought to convince the tribal leaders with whom they negotiated that they “spoke with a straight and not with a forked tongue.” According to one 1859 account, the native proverb that the “white man speaks with a forked tongue” originated as a result of the French tactic of the 1690s, in their war with the Iroquois, of inviting their enemies to attend a peace conference, only to slaughter or capture them.
And, of course, the phrase has been connected to snakes. A snake has its olfactory receptors located on its tongue. Snakes have developed forked tongues in order to detect tiny odor variations, which allows them to detect the origin of smells and track down their prey.
It doesn’t help that snakes (like spiders) generate innate feelings of disgust and (sometimes uncontrollable) fear in many mammals and most primates. So they already have a bad reputation. The parallel between the physical characteristics of the snake’s tongue and the propensity to tell lies is common in many cultures.
And, perhaps the snake’s bad reputation is reflected in our opinion of liars.