"Go fly a kite" means "get lost," but a similar expression, "to fly a kite" (or "to raise the wind") is Irish slang for writing a check without having the funds to cover it. One can imagine a check soaring farther and farther away from firm ground.
Where did the word kite come from? English poet John Milton wrote about "the wars of kites or crows flocking and fighting in the air" in his 1670 tome "History of England."
A kite, at one time Britain's most common bird of prey, was a kind of hawk with long, pointed wings. It was probably named for its cry. It would catch the wind and then glide, attracting the attention of budding toymakers.
It didn't take long for an inspired somebody to make a wooden frame over which he stretched paper and added a tail: The first recorded kite in England was made in 1664. When pulled by a string, it would rise against the wind, looking very much like the hovering bird of prey. So, naturally, they called the toy a kite.
The sense of a fictitious check appeared in 1805 in the phrase "to fly a kite." "Kiting" is now the more common usage.
It is rather doubtful that an active belfry would be the abode of bats, since these sensitive creatures would have to endure the pealing of bells.
But bats are found in abandoned towers of all kinds, including belfries that are no longer being used. The bats, which are nocturnal, slip away at night in what appears to be erratic flight. (In reality, bats navigate using a sophisticated sonar system.)
Since the belfry is the top story of a church, someone having his "top story" aflutter with disordered thoughts might be said to have "bats in his belfry."
Of course, today we call someone with crazy thoughts eccentric or deranged. However, this American folk expression has held on for almost a century, some say because of its pleasing alliteration, although nowadays we more often hear the clipped version, "batty."
SOURCES: 'Loose Cannons and Red Herrings,' by Robert Claiborne; 'Morris Dictionary of Word and Phrase Origins,' by W. and M. Morris; 'Dictionary of Clichés,' by James Rogers; 'Why You Say It,' by Webb Garrison; 'The Encyclopedia of Word and Phrase Origins,' by Robert Hendrickson.