Whoever does the most damage to en enemy aircraft gets the kill, if they're still alive. If the person who did the most is dead, it goes to the person who did the second most, ect, ect.
Damage does not mean most pieces or biggest chunks blown off, it simply means most damage. For example, each bullet has a damage rating, say:
.303 = 5
7.92mm = 6
12.7mm Breda = 25
.50 cal = 33
MG151/20 = 75
Type 99 Mk II = 85
Hispano Mk II = 100
Mk108 = 200
(It is more complicated because the impact speed affects damage amounts)
Say a P-51D and Me262 both attack a Lancaster. A Lancaster can take a lot of damagoe over the entire aircraft, it has a lot of soak. Specific "damage points" (left outer wing, left inner wing, left verticle stablizer, ect, ect) are relatively easy to destroy if you focus on them. Lets say, for the sake of an easy description, that a Lancaster has 30 damage points and each damage point can take 300 points of damage before being destroyed.
The P-51D attacks first and sprays .50 cal fire all over the Lanc, hitting 29 locations once each, only missing the left inner wing location. The Me262 then makes its attack and hits the left inner wing with three Mk108 30mm rounds. The Mk108, in my description, does 200 points per round and the Lancaster's inner wing can take 300 before being destroyed. The Me262 sees the wing come off and the Lanc tumbles down, but he only gets an assist and the P-51D gets the kill.
The Me262 hit with 600 points of damage, but only the 300 points that the Lanc's inner wing could take are counted.
The P-51D hit with 29 .50 cal rounds and each scored him 33 points for a total of 957 points. The P-51D has more than three times the damage in the Lanc, even though the P-51D didn't cause anything to break and thus he gets the kill.
Now, don't think this is a literal description of HTC's system as it isn't. They do neat stuff like have a round's energy affect its damage, calculate armor penetration, ect, ect. This is just a basic, get the gist across, version of how I understand it to work.