Here's a list of all the challenges you need to complete for each Generation in Titanfall.

2ND Generation Requirements

None, Just Reach Level 50

3RD Generation Requirements

EVA-8 Shotgun: If It Moves…
EVA-8 Shotgun: Top Gun
40mm Cannon: If It Moves…
40mm Cannon: Titan Killer

4TH Generation Requirements

R-97 Compact SMG: If It Moves…
R-97 Compact SMG: Top Gun
Plasma Railgun: If It Moves…
Plasma Railgun: Titan Killer
Satchel Charge: Top Gun

5TH Generation Requirements

Longbow-DMR Sniper: If It Moves…
Longbow-DMR Sniper: Top Gun
Quad Rocket: If It Moves…
Quad Rocket: Titan Killer
Charge Rifle: Critically Conditioned
RE-45 Autopistol: If It Moves…

6TH Generation Requirements

R-101C Carbine: If It Moves…
R-101C Carbine: Top Gun
Arc Cannon: If It Moves…
Arc Cannon: Titan Killer
Sidewinder: Titan Killer
Gooser Challenge

7TH Generation Requirements

Spitfire LMG: If It Moves…
Spitfire LMG: Top Gun
Triple Threat: If It Moves…
Triple Threat: Titan Killer
Mag Launcher: Titan Killer
Brain Surgeon

8TH Generation Requirements

G2A4 Rifle: If It Moves…
G2A4 Rifle: Top Gun
X0-16 Chaingun: If It Moves…
X0-16 Chaingun: Titan Killer
Charge Rifle: Titan Killer
Fresh Squeezed Challenge

9TH Generation Requirements

C.A.R SMG: If It Moves…
C.A.R SMG: Top Gun
Executioner
Archer Heavy Rocket: Titan Killer
All I Do is Win
Look Out Below!
40mm Cannon: Critically Conditioned

10TH Generation Requirements

Smart Pistol MK5: Top Gun
Hemlok BF-R: Top Gun
Kraber-AP Sniper: Top Gun
Disarmed
MVP
Deadly Apparition
Death Reincarnate
X0-16 Chaingun: Critically Conditioned
Plasma Railgun: Critically Conditioned