Electrical – Determining the resonant angular frequency

The circuit you're intending to use is not one of those commonly found in actual devices – exactly because the L has (nearly) zero DC resistance, so a lot of current will flow through it due to the DC bias.

It's important to notice that your (R||L||C) resonant circuit is not usually useful in this constellation in isolation. If you want to use it as the resonating part of an oscillator, you'd typically have some kind of amplifying element that feeds energy back into that oscillator; since we're talking AC here, the coupling can usually be done through a capacitor, and that means that you can isolate the biasing; you might want to think more in terms of:

^{simulate this circuit – Schematic created using CircuitLab}

More common, however, is the configuration where the bias voltage is applied over two antiparallel diodes (from German wikipedia):

Notice the choke in the bias voltage path – since the ideal voltage source has zero impedance, it would completely "swallow" any energy stuffed into your oscillator, and Lb avoids that.

Your LCR resonator has losses, so it is a decaying sinusoid wave at 5 MHz.
Before it decays to zero, it is re-excited by the negative-going edge of your 1 MHz square wave. Then re-excited by the next positive-going edge of your 1 MHz square wave....and so on.
If you wish to see more of the exponentially-decaying envelope, try changing the frequency of your generator to 714.3 kHz so that you pick out the 7th harmonic. Or try 555555 Hz to see the 9th harmonic. These will be lower amplitude of course, since the excitation energy for each harmonic is smaller.
Be aware that your oscilloscope frequency display is incorrect. It is a difficult wave to measure because of its complex shape, so perhaps frequency error is understandable. The frequency content is exactly five times your source frequency.