How is the amount of battle points earned at the end of the game counted

Everything depends on the weapons and stuff you have.

I suggest that you not use a car since you will become an easy target, but even on this I saw people winning shooting people with their car.

The best option would be to have some automatic weapon and something comparable to a sniper (or at least an AR with a x4 scope).

Turn around the blue circle toward the white circle and kill people you encounter if you can or if you can do it without being spotted too much.

The use of silencers are greatly advised.

Also, make sure you take part in people killing eachother.
For example, don't shoot at a guy being shot, but shoot the guy shooting at him.
Sometimes it's good to let him finish his target before killing him, so you don't have the target turning back on you.

The biggest word is silence and avoidng being flanked by someone.
Always be aware of what's happening around you and take the most advantageous position (something safe and with possibility to take intel from it).

Also avoid going into small cabins, since you will probably receive a grenade in there or die leaving the cabin.

As a summary:

• Silence
  
• Get information
  
• Take opportunities
  
• Place well
  
• Move when necessary (especially since gas moves a lot end game)

All depends on your decision making and your placement.

So I found a tips and tricks thread guide that explains the kind of thing I was thinking of. (A little bit down in the Control Tips section).

By default, healing items are bound to numbers 7, 8, 9, 0 and are automatically sorted in your inventory from strongest to weakest at the top of your inventory. These can be rebound as well. So from your video, the player is likely hitting one of those keys that corresponds to his healing items.