How can I electronically measure tilt/compression of a pliers or alligator-clip?
I think you want to look into small strain guages. Strain guages are a resistive element that changes resistance in relation to very small changes in the longitudinal stress introduced into the senesor element. These are generally made of a thing plastic type carrier material with loops of resistive material printed upon it. Generally a sensor is comprised of two resistive elements in series like a voltage divider. These are then wired into a bridge circuit that is monitored via an instrumentation amplifier. 

The sensors often are made up with a stick on glue to attach them to substrate. In the case of something like your allegator clip you could use two sensors just attached along the outside length of the allegator clip assembly.