Your voltmeter problem, will materialize with your wanted circuit. What you need is a very Q LC circuit and putting anything directly across it will damp the Q and the added stray capacity will de-tune it. I would build a one transistor amplifier to drive the LC circuit and optimise that circuit for output voltage and high Q- use a FET transistor to buffer the voltage to read it. So now you will have your basic LC circuit but you can drive it with a generator or micro and measure the output voltage at the buffered output.
With this sort of circuit it is usual to have an earthed shield around the coil (suitablely split so not to be a short- circuit turn), so the coil is immune to external changes of capacitance (nearer or further from the ground) and just relies on magnetic induction for any de-tuning effect.
Frank