If you are talking about your Flamboyant Lime 1964 Varsity, it would have originally come with color-matching Lime Green "flexon" vinyl tape from Schwinn (made for Schwinn by Hunt-Wilde). 1964 was the first year that Schwinn used color-matching vinyl tape on its bikes, a tradition that lasted on the electro-forged Varsity until it was discontinued in 1985. Besides being durable and looking good, it is also very easy to keep clean as unlike cotton or other tapes made from porous materials it does not absorb dirt and sweat. Note that Schwinn (and Hunt-Wilde) offered at least 3 different green flexon tapes, however there is currently some of the correct Lime tape available on eBay: Schwinn Stingray Fastback Green Handlebar Wrap Tape for Vintage Muscle Bike | eBay

Note that one 5-yard roll was used to wrap the bars, cut in half to use 2.5-yards of tape per side with a 1/8" overlap on each wind. As others have stated it was wrapped from the top-down with the ends tucked in the handlebars, which would then be capped with chrome steel "spring-type" plugs like these: Bicycle Handlebar End Plugs Schwinn Type End Plugs Caps Steel Chrome 1 Pair | eBay

These are the handlebar wrapping instructions from the Schwinn dealer service manual: