RF demodulation for dummies
A good answer might constitute a few chapters in a good DSP text, making this question overly broad.  So I'll sketch an over-simplified skeleton, where you will need to seek a decent rationale elsewhere.

Pretend your captured signal is just a vector (array of samples).

Move your signal of interest "down" to "baseband" (resulting in either 2 vectors (X,Y) or 1 complex vector (if you know complex arithmetic).
Do this by multiplying by a vector of cosines, oscillating at -8.1 MHz, to get vector X, and by a vector of sines, oscillating at the same (negative!) frequency, to get vector Y.  Low pass filter (to remove audio etc.) these 2 vectors with a cut-off somewhere around 2 or 3 MHz (see specs for your video).

The angle of a 2D vector of X,Y pairs (use atan2() to compute) will rotate with a speed proportional to the modulating signal (e.g. the video signal you want).  The speed of rotation is the first derivative of the angle with respect to time.  Or roughly, subtract successive angles, to get a time vector proportional to dA/dt.  Unwrap/rewrap phase discontinuities as needed.  Low pass filter (maybe somewhere around 2 or 3 MHz) to remove noise.  Scale as needed.  Result should be a vector containing your raw video signal (perhaps at the wrong scale, perhaps noisy, perhaps inverted, perhaps with/without gamma: fix if possible).