To weep crocodile tears is to put on an insincere show of sorrow.
The allusion is to the ancient notion that crocodiles weep while devouring their prey. Crocodiles do indeed have lachrymal glands and produce tears to lubricate the eyes as humans do. They don't cry with emotion though. Whatever experience they have when devouring prey we can be certain it isn't remorse.
There are reports of references in French that date the belief back to 1230, although I've not been able to confirm those.
The myth appeared in print in The Voyage and Travail of Sir John Maundeville, circa 1400:
"In that contre ... ben gret plentee of Cokadrilles ... Theise Serpentes slen men, and thei eten hem wepynge."
(In that country - there are many crocodiles - These serpents slay men, and then, weeping, eat them)
All of the very early citations refer directly to the myth of crocodiles weeping. It isn't until the 16th century that we find 'crocodile tears' used with our current meaning. Edmund Grindal, Archbishop of York and of Canterbury was the first to use the phrase with the implication of insincerity, in 1563, (re-published in Strype's Life of Grindal, 1711):
"I begin to fear, lest his humility ... be a counterfeit humility, and his tears crocodile tears."