David Rejected 10
Complete the street section without Goliath taking any damage in Chapter 4: The Wall
There are 8 RPG guys that are going to try and shoot the Goliath. Simply take these guys out before they can fire a shot. This can be done on any difficulty.
This starts at the "Goliath Returns" checkpoint and there may be a RPG KPA in the right corner. Credit goes to Dark_Brother. The achievement will pop right before the tank breaks through the wall. Credit goes to Stewie Griffin2
Fatal and Tragic 10
Jump off the Golden Gate Bridge in Chapter 7: Golden Gate
Go to mission select and pick 'Golden Gate'. Select the checkpoint 'Assault Down the Bridge 1'. After you take out the two RPG guys you will go past a bus on your left. There is an opening here and make sure you jump off the bridge and not the steel girder. It wasn't registering me as jumping off unless I was on the concrete and jumped into the water. Tried on two different accounts, same thing.
Some people are claiming that jumping off the steel girder does now work. If one method doesn't work, try the other.
Wilhelm's Nightmare 10
Knock 10 enemies off of the scaffolding during the helicopter fly-in in Chapter 7: Golden Gate
In the very beginning of the level where you are being flown towards the bridge, you take control of your player. Use the grenade launcher to fire at the enemies and be sure to fire a little behind them so it will knock them off the platform. They must be knocked completely off the platform for it to count.
Soft Targets 10
Destroy all vehicles using the UAV in Chapter 7: Golden Gate
Towards the end of the final mission, after the codes are rewritten for the AA guns, you will get control of the UAV. You'll be able to fire on the bridge from a drone, so simply take out the convoy of vehicles that are highlighted.
Thanks to: Flow (youtube video guides)