To unlock weapons, you should start a level where this weapon is and accomplish it with this weapon.
CMP-150: Sewer Retrieval
RCP-90: Jungle Storm
UGL Liberator: Nightclub Stakeout
DW-P5: Nightclub Stakeout
FAC-16: Trinity Escape (off dead ally)
Frag Grenade: Nightclub Stakeout
Multi Mine: Temple Surveillance
Hawk Boomerang: Rooftops Escape
KSI-74: Lab Rescue
Phychosis Gun: Trinity Infiltration
Laptop: Trinity Infiltration
Combat Sheild: Trinity Infiltraiton
M60 Mansion: Infiltration
Magnum: Rooftops Escape
Falcon: Nightclub Stakeout
Magsec 4: Mansion Infiltration
SuperDragon: Outpost Rescue
Viblade: Mansion Infiltration (red triplaser room)
DEF-12 Shotgun: Nightclub Stakeout
Rocket Launcher: Bridge Assult
Plasma Rifle: Temple Surveillance