The method which always works is to get an external mike and simply stick it inside the headset earcup.
Obviously you need a very small mike...
Then, what really helps is having a camera which has a
manual recording level control. This is quite rare as this feature is one of several used to separate the "prosumer" market from the "consumer" market.
You can make a connection directly into a headset socket, but I have no data on this. Others have done it though.
I have had excellent results (technically far better than my own narrative capability
) with my Sony HD camcorder, a little mike and a professional adjustable-gain preamp bought from the USA which clips onto the seat belt, but you don't need anything this fancy.
Whether you can connect anything to an HDMI connector I have no idea.