How should I test a coaxial cable?
You can measure a lot of things (impedance, velocity factor, distance to short-circuit, distance to open-circuit, ...) with a TDR (Time Domain Reflectometer) as shown and explained in multiple tutorials.

For example:

Cheap and simple TDR using an oscilloscope and 74AC14 Schmitt Trigger Inverter
How to measure coax velocity factor VF and impedance Z 
"TDR" or Time Domain Reflectometer, build and use this circuit
Determining Velocity Factor of coaxial cable
Understanding DTF or Distance To Fault, using a TDR
Determining Coax Impedance with a TDR