Who made the fastest car in the 60's?

It really depend in what aspect you are talking about? Production/race/limited production?

In all respects the Bill Thomas Cheetah was the fastest Top speed 215mph at le mans, 10.86@136mph, was designed as the "Cobra Killer". see link.

(production)TOP SPEED...

Ferrari 365/GTB

Lamborghini SV

Plymouth superbird

L-88 corvette

GT40

All have top speed in the low 170mph range.

quarter mile,

1968 Hurst hemi cuda-10.30@130mph

1964 Dodge light weight 330- 11.40@120

AMX/SS 11@122

Shelby cobra-12.2@118 is considered the fastest even though there are faster.

None special edition-

1968 427 corvette 12.8@112

1969 440 six pack roadrunner 12.9@111.8

1970 hemi cuda 13.1@107.12