You gain and lose "faction status" by doing stuff for, or against certain groups. Normally, completing missions for a faction will cause them to like you -- you're being "reliable". However, you can also gain faction status by doing the following:
Successfully clear a contract
Killing enemies attacking a faction's forces (must be in line of sight of faction)
Grabbing open bounties (Chinese and ROK only)
Selling a vehicle to chopshop (Mafia only)
Killing people of one faction is by far the fastest way to make them hostile towards your mercenary. If you absolutely, positively need to destroy a factions' vehicle or soldier, you should try your very best to kill only that target to minimise faction status damage. The basic rule is the more people within line of sight when you pull the trigger (or initiate the airstrike) the more you stand to lose.
If you happen to kill a faction's unit within the line of sight of five or more other faction units, you go from immediately from friendly to hostile. For purposes of accounting, "unit" means one soldier or one vehicle (despite how many soldiers are in that vehicle). Press counts as two units for purposes of calculating "units" for faction gain and loss. Having press accompanying you into a battle with DPRK soldiers will increase your faction standing with the AN, but not other factions. Other ways to lose faction status:
Destruction of a faction's vehicle (occupied) or soldier
Killing a civilian (AN, China, and ROK only)
Killing a member of the press (AN, China, and ROK only)
Destruction of key mission objectives (tents, bridges, etc)
Strangely enough, you can sometimes fool the game by doing things either out of order or by thinking outside the box. If you melee bash a soldier, you only stun them, but do not kill him. Using the take down move will reduce the status. Flashbang grenades will also not cause faction loss, but the effect is temporary. So long as there is only one soldier in the area, you can "disable" him (or her) using multiple melee bashes (or kill them with a gun) without damaging faction status. If you lose faction status when the target drops, then another unit came around when you committed your heinous act.