I think you could do that by duplicating your clip to 2 layers , grading one (or both) and applying a Blend Mode.
Trial and error ...thru' experimentation.
That looks like an extreme mid-tone contrast curve (similar to bumping the Clarity slider way up in Lightroom) with a bunch of noise on top.
Lightroom is for still photo's I believe right? Is there a simple way to do a mid-tone contrast curve in premeire or an easy to get filter? I'm pretty new at premeire :-/
High Dynamic Range is exactly what most video cameras struggle with. I think in order to do a professional looking job you'd want to be shooting RAW.
Then you could do as shooter says, expose the shot for the highlights, grade it, then reset the exposure for the blacks, grade again, and blend the two grades together using a luminosity filter, so it takes it's shadow detail from the dark exposure and it's highlight detail from the highlight one. Adjust to taste and you're done.