Here are ways to earn/unlock badges while playing online Multiplayer.

Effect	How to unlock
Ammo Expertise	Expert: Around 250 Headshots; Elite: around 500 Headshots
Armor Expertise	Expert and Elite badges are earned by finding Flak Jackets.
Explosive Expertise	Expert: Around 200 Exp. kills(MGL, SMAW or M67) Elite: 500 Exp. kills
Field Medic	Expert and Elite: HEAL!!!! Just keep on Healing!
Gadget Expertise	Expert and Elite: Get Kills with Taser, Laser Mines, Claymores
Hand-To-Hand	Expert: Around 30 Melee kills; Elite: Around 50 Melee Kills