You need to play as the Medic class to give Health Packs, and you give them by pressing Y. This goes much faster if you buy the Extra Health Pack option.
If you attempt to give a health pack and you either miss or the teammate's health is too high, the pack will fall on the ground. If a teammate walks over a dropped healthpack it will count. So the key is to hand out as many healthpacks as possible in high-traffic, high-combat areas your team controls.
On open maps (like Bank, Manor, or Tesla) you spend a lot of time chasing teammates, so the closed maps work better. In particular Hospital worked very well, sticking to the four corners on the 2nd floor, or near the stairwells.
Reviving teammates is also a good tactic, as they are immobile for a couple of seconds after getting up (and of course low on health since they were almost dead), allowing an easy health pack toss.